diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb
index 8aa037ae..1d548e07 100644
--- a/工具/批量添加题库字段数据.ipynb
+++ b/工具/批量添加题库字段数据.ipynb
@@ -2,277 +2,364 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "题号: 000960 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.941\n",
-      "题号: 002162 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.529\n",
-      "题号: 002168 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.529\n",
-      "题号: 003362 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.941\n",
-      "题号: 010602 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.794\t1.000\n",
-      "题号: 000248 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.676\n",
-      "题号: 009784 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t1.000\n",
-      "题号: 008831 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t1.000\n",
-      "题号: 000258 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.912\n",
-      "题号: 000263 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.912\n",
-      "题号: 009791 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t1.000\n",
-      "题号: 002141 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.941\n",
-      "题号: 002136 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t1.000\n",
-      "题号: 004538 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.882\n",
-      "题号: 002187 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.941\n",
-      "题号: 010622 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t1.000\t0.941\t1.000\n",
-      "题号: 008760 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.941\n",
-      "题号: 002213 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.765\n",
-      "题号: 009801 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.971\t0.912\n",
-      "题号: 009803 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.912\n",
-      "题号: 004213 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.765\n",
-      "题号: 008805 , 字段: usages 中已添加数据: 20221125\t2023届高三10班\t0.853\n",
-      "题号: 000960 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.864\n",
-      "题号: 002162 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.591\n",
-      "题号: 002168 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.682\n",
-      "题号: 003362 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.682\n",
-      "题号: 010602 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.636\t1.000\n",
-      "题号: 000248 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.773\n",
-      "题号: 009784 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.773\n",
-      "题号: 008831 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.727\n",
-      "题号: 000258 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.909\n",
-      "题号: 000263 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.727\n",
-      "题号: 009791 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.864\n",
-      "题号: 002141 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.636\n",
-      "题号: 002136 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.864\n",
-      "题号: 004538 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.909\n",
-      "题号: 002187 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.727\n",
-      "题号: 010622 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.818\t0.727\t0.864\n",
-      "题号: 008760 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.682\n",
-      "题号: 002213 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.500\n",
-      "题号: 009801 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.909\t0.864\n",
-      "题号: 009803 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.682\n",
-      "题号: 004213 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.091\n",
-      "题号: 008805 , 字段: usages 中已添加数据: 20221125\t2023届高三11班\t0.682\n",
-      "题号: 000960 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.809\n",
-      "题号: 002162 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.381\n",
-      "题号: 002168 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.905\n",
-      "题号: 003362 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.809\n",
-      "题号: 010602 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.667\t0.952\n",
-      "题号: 000248 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.429\n",
-      "题号: 009784 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t1.000\n",
-      "题号: 008831 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.905\n",
-      "题号: 000258 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.952\n",
-      "题号: 000263 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.952\n",
-      "题号: 009791 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t1.000\n",
-      "题号: 002141 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.238\n",
-      "题号: 002136 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.857\n",
-      "题号: 004538 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.809\n",
-      "题号: 002187 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.714\n",
-      "题号: 010622 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.952\t0.905\t1.000\n",
-      "题号: 008760 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.762\n",
-      "题号: 002213 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.619\n",
-      "题号: 009801 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.667\t0.667\n",
-      "题号: 009803 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.762\n",
-      "题号: 004213 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.619\n",
-      "题号: 008805 , 字段: usages 中已添加数据: 20221125\t2023届高三12班\t0.714\n",
-      "题号: 000960 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.903\n",
-      "题号: 002162 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.742\n",
-      "题号: 002168 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.774\n",
-      "题号: 003362 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.839\n",
-      "题号: 010602 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.839\t0.935\n",
-      "题号: 000248 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.581\n",
-      "题号: 009784 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.613\n",
-      "题号: 008831 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t1.000\n",
-      "题号: 000258 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.968\n",
-      "题号: 000263 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.935\n",
-      "题号: 009791 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t1.000\n",
-      "题号: 002141 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.806\n",
-      "题号: 002136 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.710\n",
-      "题号: 004538 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.903\n",
-      "题号: 002187 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.903\n",
-      "题号: 010622 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.935\t0.871\t1.000\n",
-      "题号: 008760 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.774\n",
-      "题号: 002213 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.903\n",
-      "题号: 009801 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t1.000\t0.968\n",
-      "题号: 009803 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.903\n",
-      "题号: 004213 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.774\n",
-      "题号: 008805 , 字段: usages 中已添加数据: 20221125\t2023届高三01班\t0.839\n",
-      "题号: 000960 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.700\n",
-      "题号: 002162 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.733\n",
-      "题号: 002168 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.800\n",
-      "题号: 003362 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.833\n",
-      "题号: 010602 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.933\t0.967\n",
-      "题号: 000248 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.533\n",
-      "题号: 009784 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.700\n",
-      "题号: 008831 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.833\n",
-      "题号: 000258 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.900\n",
-      "题号: 000263 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t1.000\n",
-      "题号: 009791 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.967\n",
-      "题号: 002141 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.733\n",
-      "题号: 002136 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.933\n",
-      "题号: 004538 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.833\n",
-      "题号: 002187 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.967\n",
-      "题号: 010622 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.933\t0.933\t1.000\n",
-      "题号: 008760 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.467\n",
-      "题号: 002213 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.967\n",
-      "题号: 009801 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.800\t0.833\n",
-      "题号: 009803 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.800\n",
-      "题号: 004213 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.633\n",
-      "题号: 008805 , 字段: usages 中已添加数据: 20221125\t2023届高三02班\t0.900\n",
-      "题号: 000960 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.964\n",
-      "题号: 002162 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.679\n",
-      "题号: 002168 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.679\n",
-      "题号: 003362 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.786\n",
-      "题号: 010602 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.893\t0.857\n",
-      "题号: 000248 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.500\n",
-      "题号: 009784 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.679\n",
-      "题号: 008831 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.893\n",
-      "题号: 000258 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.964\n",
-      "题号: 000263 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.964\n",
-      "题号: 009791 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.893\n",
-      "题号: 002141 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.714\n",
-      "题号: 002136 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.929\n",
-      "题号: 004538 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.893\n",
-      "题号: 002187 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.964\n",
-      "题号: 010622 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.893\t0.964\t1.000\n",
-      "题号: 008760 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.571\n",
-      "题号: 002213 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.714\n",
-      "题号: 009801 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.714\t0.607\n",
-      "题号: 009803 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.536\n",
-      "题号: 004213 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.536\n",
-      "题号: 008805 , 字段: usages 中已添加数据: 20221125\t2023届高三03班\t0.821\n",
-      "题号: 000960 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.893\n",
-      "题号: 002162 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.643\n",
-      "题号: 002168 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.571\n",
-      "题号: 003362 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.893\n",
-      "题号: 010602 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.964\t0.893\n",
-      "题号: 000248 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.607\n",
-      "题号: 009784 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.643\n",
-      "题号: 008831 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.929\n",
-      "题号: 000258 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.964\n",
-      "题号: 000263 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.821\n",
-      "题号: 009791 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t1.000\n",
-      "题号: 002141 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.607\n",
-      "题号: 002136 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.857\n",
-      "题号: 004538 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.821\n",
-      "题号: 002187 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.964\n",
-      "题号: 010622 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.964\t0.750\t0.857\n",
-      "题号: 008760 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.786\n",
-      "题号: 002213 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.679\n",
-      "题号: 009801 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.893\t0.929\n",
-      "题号: 009803 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.464\n",
-      "题号: 004213 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.607\n",
-      "题号: 008805 , 字段: usages 中已添加数据: 20221125\t2023届高三04班\t0.821\n",
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+      "题号: 009334 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 009339 , 字段: objs 中已添加数据: K0820002X\n",
+      "题号: 009343 , 字段: objs 中已添加数据: K0820001X\n",
+      "题号: 009344 , 字段: objs 中已添加数据: KNONE\n",
+      "题号: 009407 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 009408 , 字段: objs 中已添加数据: K0820002X\n",
+      "题号: 009411 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 009419 , 字段: objs 中已添加数据: K0820001X\n",
+      "题号: 009421 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 009422 , 字段: objs 中已添加数据: KNONE\n",
+      "题号: 009945 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 009946 , 字段: objs 中已添加数据: K0819006X\n",
+      "题号: 009947 , 字段: objs 中已添加数据: K0820001X\n",
+      "题号: 009948 , 字段: objs 中已添加数据: K0820003X\n",
+      "题号: 009990 , 字段: objs 中已添加数据: K0819006X\n",
+      "题号: 010875 , 字段: objs 中已添加数据: K0819002X\n",
+      "题号: 010875 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 010876 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 010877 , 字段: objs 中已添加数据: K0819002X\n",
+      "题号: 010878 , 字段: objs 中已添加数据: K0820003X\n",
+      "题号: 010879 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 010880 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 010881 , 字段: objs 中已添加数据: K0820003X\n",
+      "题号: 010882 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 010883 , 字段: objs 中已添加数据: K0819002X\n",
+      "题号: 010990 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011030 , 字段: objs 中已添加数据: KNONE\n",
+      "题号: 011054 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011137 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011226 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011269 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011293 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011308 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011340 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011347 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011369 , 字段: objs 中已添加数据: K0819002X\n",
+      "题号: 011396 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011415 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011442 , 字段: objs 中已添加数据: K0820001X\n",
+      "题号: 011459 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011498 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011528 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011628 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011651 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011703 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 011723 , 字段: objs 中已添加数据: KNONE\n",
+      "题号: 011993 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 012011 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 012026 , 字段: objs 中已添加数据: K0819005X\n",
+      "题号: 012026 , 字段: objs 中已添加数据: K0820003X\n",
+      "题号: 030022 , 字段: objs 中已有该数据: K0819005X\n",
+      "题号: 030071 , 字段: objs 中已添加数据: K0820001X\n"
      ]
     }
    ],
@@ -370,7 +457,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3.9.7 ('base')",
+   "display_name": "Python 3.8.8 ('base')",
    "language": "python",
    "name": "python3"
   },
@@ -384,12 +471,12 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.7"
+   "version": "3.8.8"
   },
   "orig_nbformat": 4,
   "vscode": {
    "interpreter": {
-    "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba"
+    "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac"
    }
   }
  },
diff --git a/工具/批量题号选题pdf生成.ipynb b/工具/批量题号选题pdf生成.ipynb
index 733f5467..c980217e 100644
--- a/工具/批量题号选题pdf生成.ipynb
+++ b/工具/批量题号选题pdf生成.ipynb
@@ -2,16 +2,16 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 1,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "开始编译教师版本pdf文件:  临时文件/抛物线预选_教师用_20221119.tex\n",
+      "开始编译教师版本pdf文件:  临时文件/二项式定理预选_教师用_20221129.tex\n",
       "0\n",
-      "开始编译学生版本pdf文件:  临时文件/抛物线预选_学生用_20221119.tex\n",
+      "开始编译学生版本pdf文件:  临时文件/二项式定理预选_学生用_20221129.tex\n",
       "0\n"
      ]
     }
@@ -26,18 +26,13 @@
     "\"\"\"---设置题目列表---\"\"\"\n",
     "#字典字段为文件名, 之后为内容的题号\n",
     "problems_dict = {\n",
-    "\"K0713001X\":\"002334,008886,008894,010671\",\n",
-    "\"K0719001X\":\"004654\",\n",
-    "\"K0719002X\":\"002393,009099\",\n",
-    "\"K0719003X\":\"000369,000467,000669,000728,000806,000833,000878,000909,000957,000968,002395,002397,002405,002406,002407,003448,003639,003837,003930,003945,004197,004221,004495,004514,004524,004550,004570,008921,008924,008926,008928,008935,008960,009077,009105,009837,009840,010682\",\n",
-    "\"K0719004X\":\"000337,000707,000739,000783,000864,002408,002409,002410,002439,003437,003449,008920,008927,008953,008956,009081,009836,010684\",\n",
-    "\"K0719005X\":\"000290,002440,002441,002450,004078,010703\",\n",
-    "\"K0719006X\":\"000275,002402,002421,002689,003440,003599,003781,008923,010683\",\n",
-    "\"K0720002X\":\"002426\",\n",
-    "\"K0720003X\":\"000282,000464,002398,002400,002401,002403,002404,002411,002412,002413,002417,002418,002419,002420,002422,002423,002424,002425,002427,002429,002430,002434,002436,002437,002445,002690,003438,003439,003441,003443,003444,003446,003447,003450,003451,004141,004225,004351,004372,004626,004639,004713,004764,008922,008925,008930,008931,008932,008933,008934,008936,008954,008955,008968,009098,009106,009110,009838,009839,010685,010688,010689\",\n",
-    "\"K0720004X\":\"000804,003795,004065,008929,008937,008957,010686\",\n",
-    "\"K0720005X\":\"002438\"\n",
-    "\n",
+    "\"K0819002X\":\"000811,002619,003997,004192,007528,007541,007544,007545,007559,007563,007574,007575,007576,007582,007604,007663,007664,010877,010883,011369\",\n",
+    "\"K0819004X\":\"000837,002621\",\n",
+    "\"K0819005X\":\"000333,000340,000373,000385,000393,000398,000410,000418,000435,000439,000470,000483,000502,000521,000532,000539,000563,000568,000580,000600,000620,000640,000658,000694,000722,000735,000737,000753,000774,000800,000823,000828,000849,000866,000873,000885,000914,000929,000951,002612,002613,002614,002615,002616,002617,002618,002620,002622,002623,002633,003584,003594,003634,003654,003735,003759,003764,003867,004021,004030,004104,004127,004148,004170,004231,004250,004298,004430,004450,004475,004517,004536,004558,004663,004686,004727,004747,007527,007529,007530,007531,007532,007542,007543,007546,007547,007549,007552,007553,007554,007555,007556,007557,007560,007561,007562,007577,007578,007581,007586,007587,007588,007589,007590,007591,007592,007593,007594,007595,007596,007597,007598,007599,007601,007602,007603,007605,007606,007607,007623,007631,007659,007660,007679,009303,009304,009305,009306,009311,009315,009316,009320,009325,009334,009407,009411,009421,009945,010875,010876,010879,010880,010882,010990,011054,011137,011226,011269,011293,011308,011340,011347,011396,011415,011459,011498,011528,011628,011651,011703,011993,012011,030022\",\n",
+    "\"K0819006X\":\"002635,002637,004028,007539,007540,007585,007628,007638,007639,007640,007641,007642,007643,007644,007645,009308,009318,009319,009946,009990\",\n",
+    "\"K0820001X\":\"002626,002627,002628,002631,002634,002639,003573,003811,003942,003962,003991,004020,004027,004211,007533,007534,007535,007536,007537,007538,007550,007564,007565,007566,007567,007568,007569,007570,007571,007572,007573,007580,007584,007611,007612,007613,007614,007615,007617,007620,007621,007625,007626,007627,007629,007630,007633,007636,007637,007646,007647,007648,007651,007658,007661,007662,007676,007677,007678,009310,009312,009314,009343,009419,009947,011442,030071\",\n",
+    "\"K0820002X\":\"000359,003840,004019,004711,007618,007622,009339,009408\",\n",
+    "\"K0820003X\":\"002624,002629,002630,002636,003578,003851,003883,004342,004625,004678,007548,007551,007558,007608,009313,009948,010878,010881,012026\",\n",
     "}\n",
     "\n",
     "\"\"\"---设置题目列表结束---\"\"\"\n",
@@ -45,7 +40,7 @@
     "\"\"\"---设置文件保存路径---\"\"\"\n",
     "#目录和文件的分隔务必用/\n",
     "directory = \"临时文件/\"\n",
-    "filename = \"抛物线预选\"\n",
+    "filename = \"二项式定理预选\"\n",
     "\"\"\"---设置文件名结束---\"\"\"\n",
     "if directory[-1] != \"/\":\n",
     "    directory += \"/\"\n",
diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt
index a7a751f9..bf4a835a 100644
--- a/工具/文本文件/metadata.txt
+++ b/工具/文本文件/metadata.txt
@@ -1,794 +1,1362 @@
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+9334
+K0819005X
+
+
+9339
+K0820002X
+
+
+9343
+K0820001X
+
+
+9344
+KNONE
+
+
+9407
+K0819005X
+
+
+9408
+K0820002X
+
+
+9411
+K0819005X
+
+
+9419
+K0820001X
+
+
+9421
+K0819005X
+
+
+9422
+KNONE
+
+
+9945
+K0819005X
+
+
+9946
+K0819006X
+
+
+9947
+K0820001X
+
+
+9948
+K0820003X
+
+
+9990
+K0819006X
+
+
+10875
+K0819002X
+K0819005X
+
+
+10876
+K0819005X
+
+
+10877
+K0819002X
+
+
+10878
+K0820003X
+
+
+10879
+K0819005X
+
+
+10880
+K0819005X
+
+
+10881
+K0820003X
+
+
+10882
+K0819005X
+
+
+10883
+K0819002X
+
+
+10990
+K0819005X
+
+
+11030
+KNONE
+
+
+11054
+K0819005X
+
+
+11137
+K0819005X
+
+
+11226
+K0819005X
+
+
+11269
+K0819005X
+
+
+11293
+K0819005X
+
+
+11308
+K0819005X
+
+
+11340
+K0819005X
+
+
+11347
+K0819005X
+
+
+11369
+K0819002X
+
+
+11396
+K0819005X
+
+
+11415
+K0819005X
+
+
+11442
+K0820001X
+
+
+11459
+K0819005X
+
+
+11498
+K0819005X
+
+
+11528
+K0819005X
+
+
+11628
+K0819005X
+
+
+11651
+K0819005X
+
+
+11703
+K0819005X
+
+
+11723
+KNONE
+
+
+11993
+K0819005X
+
+
+12011
+K0819005X
+
+
+12026
+K0819005X
+K0820003X
+
+
+30022
+K0819005X
+
+
+30071
+K0820001X
+
 
-008805
-20221125	2023届高三09班	0.700
 
diff --git a/工具/根据目标列表批量生成对应题目的字典.ipynb b/工具/根据目标列表批量生成对应题目的字典.ipynb
index 500863b5..57ee1fa1 100644
--- a/工具/根据目标列表批量生成对应题目的字典.ipynb
+++ b/工具/根据目标列表批量生成对应题目的字典.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 1,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -14,44 +14,39 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [],
    "source": [
     "# 输入目标列表\n",
-    "t = \"\"\"K0719001X\n",
-    "K0719002X\n",
-    "K0719003X\n",
-    "K0719004X\n",
-    "K0719005X\n",
-    "K0719006X\n",
-    "K0720001X\n",
-    "K0720002X\n",
-    "K0720003X\n",
-    "K0720004X\n",
-    "K0720005X\n",
+    "t = \"\"\"K0819001X\n",
+    "K0819002X\n",
+    "K0819003X\n",
+    "K0819004X\n",
+    "K0819005X\n",
+    "K0819006X\n",
+    "K0820001X\n",
+    "K0820002X\n",
+    "K0820003X\n",
     "\"\"\""
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "\"K0719001X\":\"004654\",\n",
-      "\"K0719002X\":\"002393,009099\",\n",
-      "\"K0719003X\":\"000369,000467,000669,000728,000806,000833,000878,000909,000957,000968,002395,002397,002405,002406,002407,003448,003639,003837,003930,003945,004197,004221,004495,004514,004524,004550,004570,008921,008924,008926,008928,008935,008960,009077,009105,009837,009840,010682\",\n",
-      "\"K0719004X\":\"000337,000707,000739,000783,000864,002408,002409,002410,002439,003437,003449,008920,008927,008953,008956,009081,009836,010684\",\n",
-      "\"K0719005X\":\"000290,002440,002441,002450,004078,010703\",\n",
-      "\"K0719006X\":\"000275,002402,002421,002689,003440,003599,003781,008923,010683\",\n",
-      "\"K0720002X\":\"002426\",\n",
-      "\"K0720003X\":\"000282,000464,002398,002400,002401,002403,002404,002411,002412,002413,002417,002418,002419,002420,002422,002423,002424,002425,002427,002429,002430,002434,002436,002437,002445,002690,003438,003439,003441,003443,003444,003446,003447,003450,003451,004141,004225,004351,004372,004626,004639,004713,004764,008922,008925,008930,008931,008932,008933,008934,008936,008954,008955,008968,009098,009106,009110,009838,009839,010685,010688,010689\",\n",
-      "\"K0720004X\":\"000804,003795,004065,008929,008937,008957,010686\",\n",
-      "\"K0720005X\":\"002438\",\n"
+      "\"K0819002X\":\"000811,002619,003997,004192,007528,007541,007544,007545,007559,007563,007574,007575,007576,007582,007604,007663,007664,010877,010883,011369\",\n",
+      "\"K0819004X\":\"000837,002621\",\n",
+      "\"K0819005X\":\"000333,000340,000373,000385,000393,000398,000410,000418,000435,000439,000470,000483,000502,000521,000532,000539,000563,000568,000580,000600,000620,000640,000658,000694,000722,000735,000737,000753,000774,000800,000823,000828,000849,000866,000873,000885,000914,000929,000951,002612,002613,002614,002615,002616,002617,002618,002620,002622,002623,002633,003584,003594,003634,003654,003735,003759,003764,003867,004021,004030,004104,004127,004148,004170,004231,004250,004298,004430,004450,004475,004517,004536,004558,004663,004686,004727,004747,007527,007529,007530,007531,007532,007542,007543,007546,007547,007549,007552,007553,007554,007555,007556,007557,007560,007561,007562,007577,007578,007581,007586,007587,007588,007589,007590,007591,007592,007593,007594,007595,007596,007597,007598,007599,007601,007602,007603,007605,007606,007607,007623,007631,007659,007660,007679,009303,009304,009305,009306,009311,009315,009316,009320,009325,009334,009407,009411,009421,009945,010875,010876,010879,010880,010882,010990,011054,011137,011226,011269,011293,011308,011340,011347,011396,011415,011459,011498,011528,011628,011651,011703,011993,012011,030022\",\n",
+      "\"K0819006X\":\"002635,002637,004028,007539,007540,007585,007628,007638,007639,007640,007641,007642,007643,007644,007645,009308,009318,009319,009946,009990\",\n",
+      "\"K0820001X\":\"002626,002627,002628,002631,002634,002639,003573,003811,003942,003962,003991,004020,004027,004211,007533,007534,007535,007536,007537,007538,007550,007564,007565,007566,007567,007568,007569,007570,007571,007572,007573,007580,007584,007611,007612,007613,007614,007615,007617,007620,007621,007625,007626,007627,007629,007630,007633,007636,007637,007646,007647,007648,007651,007658,007661,007662,007676,007677,007678,009310,009312,009314,009343,009419,009947,011442,030071\",\n",
+      "\"K0820002X\":\"000359,003840,004019,004711,007618,007622,009339,009408\",\n",
+      "\"K0820003X\":\"002624,002629,002630,002636,003578,003851,003883,004342,004625,004678,007548,007551,007558,007608,009313,009948,010878,010881,012026\",\n"
      ]
     }
    ],
diff --git a/工具/目标挂钩简要清点.ipynb b/工具/目标挂钩简要清点.ipynb
index e58c0688..2109745c 100644
--- a/工具/目标挂钩简要清点.ipynb
+++ b/工具/目标挂钩简要清点.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -24,22 +24,22 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "第一单元 . 总题数: 1766 , 完成对应题数: 1400\n",
+      "第一单元 . 总题数: 1767 , 完成对应题数: 1400\n",
       "第二单元 . 总题数: 2495 , 完成对应题数: 2127\n",
       "第三单元 . 总题数: 2079 , 完成对应题数: 395\n",
-      "第四单元 . 总题数: 1126 , 完成对应题数: 1032\n",
-      "第五单元 . 总题数: 1329 , 完成对应题数: 305\n",
+      "第四单元 . 总题数: 1127 , 完成对应题数: 1033\n",
+      "第五单元 . 总题数: 1329 , 完成对应题数: 306\n",
       "第六单元 . 总题数: 1074 , 完成对应题数: 431\n",
-      "第七单元 . 总题数: 1338 , 完成对应题数: 953\n",
-      "第八单元 . 总题数: 1221 , 完成对应题数: 195\n",
-      "第九单元 . 总题数: 252 , 完成对应题数: 101\n"
+      "第七单元 . 总题数: 1342 , 完成对应题数: 957\n",
+      "第八单元 . 总题数: 1194 , 完成对应题数: 519\n",
+      "第九单元 . 总题数: 277 , 完成对应题数: 101\n"
      ]
     }
    ],
@@ -50,16 +50,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "(12680, 6939)"
+       "(12684, 7269)"
       ]
      },
-     "execution_count": 3,
+     "execution_count": 8,
      "metadata": {},
      "output_type": "execute_result"
     }
diff --git a/工具/讲义生成.ipynb b/工具/讲义生成.ipynb
index 3dfb7e1c..10546d20 100644
--- a/工具/讲义生成.ipynb
+++ b/工具/讲义生成.ipynb
@@ -13,9 +13,9 @@
       "题块 1 处理完毕.\n",
       "正在处理题块 2 .\n",
       "题块 2 处理完毕.\n",
-      "开始编译教师版本pdf文件:  临时文件/37_抛物线的概念及性质_教师_20221129.tex\n",
+      "开始编译教师版本pdf文件:  临时文件/38_计数原理与排列组合_教师_20221129.tex\n",
       "0\n",
-      "开始编译学生版本pdf文件:  临时文件/37_抛物线的概念及性质_学生_20221129.tex\n",
+      "开始编译学生版本pdf文件:  临时文件/38_计数原理与排列组合_学生_20221129.tex\n",
       "0\n"
      ]
     }
@@ -35,7 +35,7 @@
     "\n",
     "\"\"\"---设置其他预处理替换命令---\"\"\"\n",
     "#2023届第一轮讲义更换标题\n",
-    "exec_list = [(\"标题数字待处理\",\"37\"),(\"标题文字待处理\",\"抛物线的概念及性质\")] \n",
+    "exec_list = [(\"标题数字待处理\",\"38\"),(\"标题文字待处理\",\"计数原理与排列组合\")] \n",
     "enumi_mode = 0\n",
     "\n",
     "#2023届测验卷与周末卷\n",
@@ -49,14 +49,14 @@
     "\"\"\"---其他预处理替换命令结束---\"\"\"\n",
     "\n",
     "\"\"\"---设置目标文件名---\"\"\"\n",
-    "destination_file = \"临时文件/37_抛物线的概念及性质\"\n",
+    "destination_file = \"临时文件/38_计数原理与排列组合\"\n",
     "\"\"\"---设置目标文件名结束---\"\"\"\n",
     "\n",
     "\n",
     "\"\"\"---设置题号数据---\"\"\"\n",
     "problems = [\n",
-    "\"669,3837,4495,3437,2402,2426,2403,2417,804,282,2689,4550\",\n",
-    "\",8926,3438,3781,9081,2424,2418,8923,2434,8956,2429,8937\"\n",
+    "\"7366,7371,7376,7368,7501,9254,2605,3567,2602,3581,7423,9274,2550,9935,9285,7361,7506,10868,11229,7447,7450,11679,9355\",\n",
+    "\"7380,7673,7455,7502,3666,2586,2537,7432,2563,2549,9262,7521,9333,10861,7494,7476,3640,9351,3574\"\n",
     "\n",
     "]\n",
     "\"\"\"---设置题号数据结束---\"\"\"\n",
@@ -208,7 +208,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3.9.7 ('base')",
+   "display_name": "Python 3.8.8 ('base')",
    "language": "python",
    "name": "python3"
   },
@@ -222,12 +222,12 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.7"
+   "version": "3.8.8"
   },
   "orig_nbformat": 4,
   "vscode": {
    "interpreter": {
-    "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba"
+    "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac"
    }
   }
  },
diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb
index fbd14b4c..c5d3574b 100644
--- a/工具/题号选题pdf生成.ipynb
+++ b/工具/题号选题pdf生成.ipynb
@@ -2,16 +2,16 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 1,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "开始编译教师版本pdf文件:  临时文件/提高03_教师用_20221129.tex\n",
+      "开始编译教师版本pdf文件:  临时文件/题库_教师用_20221129.tex\n",
       "0\n",
-      "开始编译学生版本pdf文件:  临时文件/提高03_学生用_20221129.tex\n",
+      "开始编译学生版本pdf文件:  临时文件/题库_学生用_20221129.tex\n",
       "0\n"
      ]
     }
@@ -26,14 +26,14 @@
     "\"\"\"---设置题目列表---\"\"\"\n",
     "#留空为编译全题库, a为读取临时文件中的题号筛选.txt文件生成题库\n",
     "problems = r\"\"\"\n",
-    "30485:30493\n",
+    "\n",
     "\n",
     "\"\"\"\n",
     "\"\"\"---设置题目列表结束---\"\"\"\n",
     "\n",
     "\"\"\"---设置文件名---\"\"\"\n",
     "#目录和文件的分隔务必用/\n",
-    "filename = \"临时文件/提高03\"\n",
+    "filename = \"临时文件/题库\"\n",
     "\"\"\"---设置文件名结束---\"\"\"\n",
     "\n",
     "\n",
@@ -174,7 +174,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3.9.7 ('base')",
+   "display_name": "Python 3.8.8 ('base')",
    "language": "python",
    "name": "python3"
   },
@@ -188,12 +188,12 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.7"
+   "version": "3.8.8"
   },
   "orig_nbformat": 4,
   "vscode": {
    "interpreter": {
-    "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba"
+    "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac"
    }
   }
  },
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index 96e84dd2..3af29600 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -8849,7 +8849,8 @@
         "content": "设常数$a>0$, $(x+\\dfrac{a}{\\sqrt{x}})^9$展开式中$x^6$的系数为$4$, 则$\\displaystyle\\lim_{n\\to \\infty}(a+a^2+\\cdots+a^n)=$\\blank{50}.",
         "objs": [
             "K0405002X",
-            "K0405003X"
+            "K0405003X",
+            "K0819005X"
         ],
         "tags": [
             "第四单元",
@@ -9087,7 +9088,9 @@
     "000340": {
         "id": "000340",
         "content": "在$(x+\\dfrac{2}{x^2})^6$的二项展开式中第四项的系数是\\blank{50}(结果用数值表示).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -9462,7 +9465,9 @@
     "000350": {
         "id": "000350",
         "content": "已知$(a+3b)^n$的展开式中, 各项系数的和与各项二项式系数的和之比为$64$, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -9760,7 +9765,9 @@
     "000359": {
         "id": "000359",
         "content": "若$(1+x)^5=a_0+a_1x+a_2x^2+\\cdots+a_5x^5$, 则$a_1+a_2+\\cdots+a_5=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -10279,7 +10286,9 @@
     "000373": {
         "id": "000373",
         "content": "若$(2x^2+\\dfrac1x)^n$($n\\in \\mathbf{N}$, $n\\ge 1$)的二项展开式中的第$9$项是常数项, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -10720,7 +10729,9 @@
     "000385": {
         "id": "000385",
         "content": "设常数$a>0$, 若$(x+\\dfrac ax)^9$的二项展开式中$x^5$的系数为$144$, 则$a=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -11030,7 +11041,9 @@
     "000393": {
         "id": "000393",
         "content": "设$(1+x)^n=a_0+a_1x+a_2x^2+a_3x^3+\\cdots +a_nx^n$, 若$\\dfrac{a_2}{a_3}=\\dfrac13$, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -11199,7 +11212,9 @@
     "000398": {
         "id": "000398",
         "content": "在二项式$(x+\\dfrac2x)^6$的展开式中, 常数项是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -11532,7 +11547,9 @@
     "000410": {
         "id": "000410",
         "content": "$(1+2x)^6$展开式中$x^3$项的系数为\\blank{50}(用数字作答).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -11753,7 +11770,9 @@
     "000418": {
         "id": "000418",
         "content": "$(1-\\dfrac x2)^8$的二项展开式中含$x^2$项的系数是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -12215,7 +12234,9 @@
     "000435": {
         "id": "000435",
         "content": "设函数$f(x)=\\begin{cases}    x^6, & x\\ge 1,  \\\\   -2x-1, & x\\le -1, \\end{cases}$ 则当$x\\le -1$时, $f[f(x)]$表达式的展开式中含${x^2}$项的系数是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -12317,7 +12338,9 @@
     "000439": {
         "id": "000439",
         "content": "二项式$(x^2+\\dfrac 1x)^5$的展开式中, $x$的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -12927,7 +12950,9 @@
     "000462": {
         "id": "000462",
         "content": "在$(\\dfrac3{x^2}+\\sqrt{x})^n$的二项展开式中, 所有项的二项式系数之和为$1024$, 则常数项的值等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -13149,7 +13174,9 @@
     "000470": {
         "id": "000470",
         "content": "在代数式$(x+\\dfrac 1{x^2})^7$的展开式中, 一次项的系数是\\blank{50}(用数字作答).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -13501,7 +13528,9 @@
     "000483": {
         "id": "000483",
         "content": "$(x+\\dfrac{1}{x^2})^9$的二项展开式中, 常数项的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -14004,7 +14033,9 @@
     "000502": {
         "id": "000502",
         "content": "已知二项式展开式$(1-2x)^7=a_0+a_1x+a_2x^2+\\cdots +a_7x^7$, 且复数$z=\\dfrac12a_1+\\dfrac{a_7}{128}\\mathrm{i}$, 则复数$z$的模$|z|=$\\blank{50}(其中$\\mathrm{i}$是虚数单位).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第五单元",
             "第八单元",
@@ -14546,7 +14577,9 @@
     "000521": {
         "id": "000521",
         "content": "在$(x-a)^{10}$的展开式中, $x^7$的系数是$15$, 则实数$a=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -14827,7 +14860,9 @@
     "000532": {
         "id": "000532",
         "content": "在$(1+2x)^5$的展开式中, $x^2$项系数为\\blank{50}(用数字作答).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -15003,7 +15038,9 @@
     "000539": {
         "id": "000539",
         "content": "$(\\sqrt x-\\dfrac1x)^9$的二项展开式中的常数项的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -15644,7 +15681,9 @@
     "000563": {
         "id": "000563",
         "content": "已知$(1+2x)^6$展开式的二项式系数的最大值为$a$, 系数的最大值为$b$, 则$\\dfrac ba=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -15778,7 +15817,9 @@
     "000568": {
         "id": "000568",
         "content": "二项式$(x-\\dfrac1{2x})^4$的展开式中的常数项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -16106,7 +16147,9 @@
     "000580": {
         "id": "000580",
         "content": "在$(x-\\dfrac2x)^6$的二项展开式中, 常数项的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -16462,7 +16505,9 @@
     "000593": {
         "id": "000593",
         "content": "若$(2x+\\dfrac 1x)^n$的二项展开式中的所有二项式系数之和等于$256$, 则该展开式中常数项的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -17162,7 +17207,9 @@
     "000620": {
         "id": "000620",
         "content": "若$(x+2)^n=x^n+ax^{n-1}+\\cdots+bx+c \\ (n\\in \\mathbf{N}^*, \\ n\\ge 3)$, 且$b=4c$, 则$a$的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -17489,7 +17536,9 @@
     "000632": {
         "id": "000632",
         "content": "若$(\\sqrt x-\\dfrac1x)^n$的二项展开式中各项的二项式系数的和是$64$, 则展开式中的常数项的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -17719,7 +17768,9 @@
     "000640": {
         "id": "000640",
         "content": "若$(x+a)^7$的二项展开式中, 含$x^6$项的系数为$7$, 则实数$a=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -18197,7 +18248,9 @@
     "000658": {
         "id": "000658",
         "content": "已知$(x-\\dfrac ax)^7$展开式中$x^3$的系数为$84$, 则正实数$a$的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -19177,7 +19230,9 @@
     "000694": {
         "id": "000694",
         "content": "设多项式$1+x+(1+x)^2+(1+x)^3+\\cdots+(1+x)^n\\ (x\\ne 0, \\ n\\in \\mathbf{N}^*)$的展开式中$x$项的系数为$T_n$, 则$\\displaystyle\\lim_{n\\to \\infty}\\dfrac{T_n}{n^2}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -19934,7 +19989,9 @@
     "000722": {
         "id": "000722",
         "content": "若二项式$(2x+\\dfrac ax)^7$的展开式中一次项的系数是$-70$, 则$\\displaystyle\\lim_{n\\to\\infty}(a+a^2+a^3+\\cdots+a^n)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -20283,7 +20340,9 @@
     "000735": {
         "id": "000735",
         "content": "代数式$(x^2+2)(\\dfrac1{x^2}-1)^5$的展开式的常数项是\\blank{50}(用数字作答).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -20333,7 +20392,9 @@
     "000737": {
         "id": "000737",
         "content": "在$(x+\\dfrac1x)^6$的二项展开式中, 常数项是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -20754,7 +20815,9 @@
     "000753": {
         "id": "000753",
         "content": "若将函数$f(x)=x^6$表示成$f(x)=a_0+a_1(x-1)+a_2(x-1)^2+a_3(x-1)^3+\\cdots+a_6(x-1)^6$则$a_3$的值等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -21329,7 +21392,9 @@
     "000774": {
         "id": "000774",
         "content": "$(1+2x)^n$的二项展开式中, 含$x^3$项的系数等于含$x$项的系数的$8$倍, 则正整数$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -22060,7 +22125,9 @@
     "000800": {
         "id": "000800",
         "content": "$(\\sqrt x+\\dfrac1x)^9$二项展开式中的常数项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -22347,7 +22414,9 @@
     "000811": {
         "id": "000811",
         "content": "若$(x^3-\\dfrac1{x^2})^n$的展开式中含有非零常数项, 则正整数$n$的最小值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -22672,7 +22741,9 @@
     "000823": {
         "id": "000823",
         "content": "$(1+\\dfrac1{x^2})(1+x)^6$展开式中$x^2$的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -22821,7 +22892,9 @@
     "000828": {
         "id": "000828",
         "content": "若$(1+3x)^n$的二项展开式中$x^2$项的系数是$54$, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -23053,7 +23126,9 @@
     "000837": {
         "id": "000837",
         "content": "$(x+\\dfrac1x)^n$的展开式中的第$3$项为常数项, 则正整数$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819004X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -23373,7 +23448,9 @@
     "000849": {
         "id": "000849",
         "content": "已知$(ax+\\dfrac1x)^6$二项展开式中的第五项系数为$\\dfrac{15}2$, 则正实数$a$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -23813,7 +23890,9 @@
     "000866": {
         "id": "000866",
         "content": "记$(2x+\\dfrac1x)^n \\ (n\\in \\mathbf{N}^*$)的展开式中第$m$项的系数为$b_m$, 若$b_3=2b_4$, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -24011,7 +24090,9 @@
     "000873": {
         "id": "000873",
         "content": "$(\\dfrac1x-\\sqrt x)^6$的展开式中, 常数项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -24344,7 +24425,9 @@
     "000885": {
         "id": "000885",
         "content": "试写出$(x-\\dfrac1x)^7$展开式中系数最大的项\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -24902,7 +24985,9 @@
     "000906": {
         "id": "000906",
         "content": "若二项式$(2x-\\dfrac1{\\sqrt x})^n$展开式中的第$5$项为常数项, 则展开式中各项的二项式系数之和为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -25105,7 +25190,9 @@
     "000914": {
         "id": "000914",
         "content": "在$(x^3-\\dfrac1x)^8$的展开式中, 其常数项的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -25484,7 +25571,9 @@
     "000929": {
         "id": "000929",
         "content": "设$a\\ne 0$, $n$是大于$1$的自然数, $(1+\\dfrac xa)^n$的展开式为$a_0+a_1x+a_2x^2+\\cdots+a_nx^n$. 若$a_1=3$, $a_2=4$, 则$a=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -26069,7 +26158,9 @@
     "000951": {
         "id": "000951",
         "content": "在代数式$(4x^2-2x-5)(1+\\dfrac1{x^2})^5$的展开式中, 常数等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70675,7 +70766,9 @@
     "002611": {
         "id": "002611",
         "content": "$(x-2y)^{10}$的展开式中, 第$4$项的二项式系数为\\blank{80}, 第$4$项的系数为\\blank{80}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70700,7 +70793,9 @@
     "002612": {
         "id": "002612",
         "content": "$\\left(\\dfrac{1}{\\sqrt{x}}-\\sqrt[3]{x}\\right)^{20}$的展开式中, 不含$x$的项是\\blank{80}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70725,7 +70820,9 @@
     "002613": {
         "id": "002613",
         "content": "$\\left(x+2x^{-2}\\right)^n$的展开式中, 第三项为常数, 则中间项为\\blank{80}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70750,7 +70847,9 @@
     "002614": {
         "id": "002614",
         "content": "若$\\left(\\sqrt[3]{x}-\\dfrac{2}{x}\\right)^{n}$的展开式中, 第$8$项含$x^{\\frac{1}{3}}$,\n则含$x^{-1}$的是第\\blank{80}项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70775,7 +70874,9 @@
     "002615": {
         "id": "002615",
         "content": "在$\\left(\\sqrt{2}+\\sqrt[3]{3}\\right)^{100}$的展开式中, 有\\blank{80}项是有理项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70800,7 +70901,9 @@
     "002616": {
         "id": "002616",
         "content": "$\\left(x+x^{-1}-1\\right)^5$中的常数项为\\blank{80}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70825,7 +70928,9 @@
     "002617": {
         "id": "002617",
         "content": "在$(ax+1)^7$的展开式中, 已知$x^3$的系数是$x^2$的系数与$x^4$的系数的等差中项, 且实数$a>1$, 则$a=$\\blank{30}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70852,7 +70957,9 @@
     "002618": {
         "id": "002618",
         "content": "$(1+x)+(1+x)^2+(1+x)^3+\\cdots+(1+x)^{2n} \\ (n\\in \\mathbf{Z}^+)$的展开式中$x^n$项的系数为\\blank{30}(用单个组合数表示).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70879,7 +70986,9 @@
     "002619": {
         "id": "002619",
         "content": "利用$(1+t)^5$的展开式化简: $(2x+1)^5-5(2x+1)^4+10(2x+1)^3-10(2x+1)^2+5(2x+1)-1=$\\blank{80}.",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70904,7 +71013,9 @@
     "002620": {
         "id": "002620",
         "content": "$(x+2y+z)^9$的展开式中含$x^2y^3z^4$项的系数为\\blank{80}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70931,7 +71042,10 @@
     "002621": {
         "id": "002621",
         "content": "[选做]\n设$n$是正整数, 将$(1+x+x^2)^n$的展开式中, $x^k$的系数记作$\\mathrm{D}_n^k$, 称为三项式系数.\n参照二项式系数杨辉三角形中的规律, 探究三项式系数的规律. 要求写出相应的三角数阵, 证明并描述所得数阵的规律.",
-        "objs": [],
+        "objs": [
+            "K0819002X",
+            "K0819004X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70956,7 +71070,9 @@
     "002622": {
         "id": "002622",
         "content": "$(1+\\mathrm{i})^{10}$的展开式中, 第八项为\\blank{80}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -70981,7 +71097,9 @@
     "002623": {
         "id": "002623",
         "content": "在$(1+x)^n$的展开式中, 若第三项和第六项的系数相等, 则$n=$\\blank{80}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71006,7 +71124,9 @@
     "002624": {
         "id": "002624",
         "content": "二项式$(x-y)^{99}$中, 系数最小的项是第\\blank{40}项, 系数最大的项是第\\blank{40}项.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71031,7 +71151,9 @@
     "002625": {
         "id": "002625",
         "content": "$(2-3x)^n$展开式中各项系数之和为\\blank{40}, 各项二项式系数之和为\\blank{40},\n$x$的奇次幂的项的系数之和为\\blank{40}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71056,7 +71178,9 @@
     "002626": {
         "id": "002626",
         "content": "$(2x-1)^5$的展开式中, 各项系数的绝对值之和为\\blank{80}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71083,7 +71207,9 @@
     "002627": {
         "id": "002627",
         "content": "当$n$是正整数时, $1-2\\mathrm{C}_n^1+4\\mathrm{C}_n^2-8\\mathrm{C}_n^3+\\cdots+(-2)^n\\mathrm{C}_n^n=$\\blank{80}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71108,7 +71234,9 @@
     "002628": {
         "id": "002628",
         "content": "求值: $\\mathrm{C}_{100}^{0}-\\mathrm{C}_{100}^2+\\mathrm{C}_{100}^4-\\mathrm{C}_{100}^6+\\cdots-\\mathrm{C}_{100}^{98}+\\mathrm{C}_{100}^{100}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71133,7 +71261,9 @@
     "002629": {
         "id": "002629",
         "content": "求$(2x+3y)^{11}$的展开式中系数最大的项.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71158,7 +71288,10 @@
     "002630": {
         "id": "002630",
         "content": "在$(1+x)^n$的展开式中, 有连续三项的二项式系数之比为$3:8:14$, 试在展开式的所有项中, 求系数最大的项.",
-        "objs": [],
+        "objs": [
+            "K0819005X",
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71183,7 +71316,9 @@
     "002631": {
         "id": "002631",
         "content": "(1) 求证: $k\\mathrm{C}_n^k=n\\mathrm{C}_{n-1}^{k-1}$.\\\\ \n(2) {\\it (选做)}已知$n$是正整数, 求$\\mathrm{C}_n^0+\\dfrac{1}{2}\\mathrm{C}_n^1+\\dfrac{1}{3}\\mathrm{C}_n^2+\\cdots+\\dfrac{1}{n+1}\\mathrm{C}_n^n$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理",
@@ -71234,7 +71369,9 @@
     "002633": {
         "id": "002633",
         "content": "$(a+b+c)^5$合并同类项后共有\\blank{40}项, 其中$a^3bc$的系数为\\blank{40}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71259,7 +71396,10 @@
     "002634": {
         "id": "002634",
         "content": "$(1+2x+x^2)^{10}(1-x)^5$的展开式中, 各项系数的和为\\blank{30}, 常数项为\\blank{30},\n$x$的奇次项系数的和为\\blank{30}.",
-        "objs": [],
+        "objs": [
+            "K0820001X",
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71284,7 +71424,9 @@
     "002635": {
         "id": "002635",
         "content": "$77^{77}-15$除以$19$的余数为\\blank{40}.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71311,7 +71453,9 @@
     "002636": {
         "id": "002636",
         "content": "$\\left(1+\\sqrt{2}\\right)^{50}$的展开式中最大的项为\\blank{100}.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71336,7 +71480,9 @@
     "002637": {
         "id": "002637",
         "content": "证明: 无论$n$是何正整数, $(n+1)^n-1$能被$n^2$整除.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -71386,7 +71532,9 @@
     "002639": {
         "id": "002639",
         "content": "[选做]\n利用复数的三角形式的有关性质及二项式定理证明:\\\\ \n(1) $1+\\mathrm{C}_n^1\\cos\\alpha+\\mathrm{C}_n^2\\cos 2\\alpha+\\mathrm{C}_n^3\\cos 3\\alpha+\\cdots+\\mathrm{C}_n^n\\cos n\\alpha=2^n\\cos^n\\dfrac{\\alpha}{2}\\cos\\dfrac{n\\alpha}{2}$;\\\\ \n(2) $\\mathrm{C}_n^1\\sin\\alpha+\\mathrm{C}_n^2\\sin 2\\alpha+\\mathrm{C}_n^3\\sin 3\\alpha+\\cdots+\\mathrm{C}_n^n\\sin n\\alpha=2^n\\cos^n\\dfrac{\\alpha}{2}\\sin\\dfrac{n\\alpha}{2}$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -95158,7 +95306,9 @@
     "003572": {
         "id": "003572",
         "content": "$(2x-\\dfrac 1{\\sqrt x})^6$的展开式中第三项的二项式系数为\\blank{50}, 第三项的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -95180,7 +95330,9 @@
     "003573": {
         "id": "003573",
         "content": "化简: (1) $1+2\\mathrm{C}_n^1+4\\mathrm{C}_n^2+\\cdots+2^n\\mathrm{C}_n^n=$\\blank{50};\\\\\n(2) $\\mathrm{C}_3^3+\\mathrm{C}_4^3+\\mathrm{C}_5^3+\\cdots+\\mathrm{C}_n^3=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -95291,7 +95443,10 @@
     "003578": {
         "id": "003578",
         "content": "求二项式$(2+x^2)^{32}$展开式中,\\\\\n(1) 二项式系数最大的项;\\\\\n(2) 系数最大的项;\\\\\n(3) 所有项的系数之和.",
-        "objs": [],
+        "objs": [
+            "K0820001X",
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -95403,7 +95558,9 @@
     "003583": {
         "id": "003583",
         "content": "在$(\\sqrt[3]x-\\dfrac 1{2\\cdot \\sqrt[3]x})^{10}$的二项展开式中, 常数项是\\blank{50}; 含$x^2$项的二项式系数是\\blank{50}; 含$x^2$项的系数是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -95425,7 +95582,9 @@
     "003584": {
         "id": "003584",
         "content": "设$1+(1+x)+(1+x)^2+\\cdots+(1+x)^{2021}=a_0+a_1x+a_2x^2+\\cdots +a_{2021}x^{2021}$, 则$a_{100}+a_{101}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -95658,7 +95817,9 @@
     "003594": {
         "id": "003594",
         "content": "已知二项式$(x+a)^5$展开式中, $x^2$项的系数为$80$, 则$a=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -96687,7 +96848,9 @@
     "003634": {
         "id": "003634",
         "content": "在二项式$(2x+1)^5$的展开式中, $x^2$的系数是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -97193,7 +97356,9 @@
     "003654": {
         "id": "003654",
         "content": "在$(1+x)^7$的二项展开式中, $x^2$项的系数为\\blank{50}(结果用数值表示).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -99200,7 +99365,9 @@
     "003735": {
         "id": "003735",
         "content": "若二项式$\\left(ax-\\dfrac{\\sqrt{3}}{6}\\right)^3$的展开式的第二项系数为$-\\dfrac{\\sqrt{3}}{2}$, 则实数$a$的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -99549,7 +99716,9 @@
     "003750": {
         "id": "003750",
         "content": "已知$\\left(\\sqrt{x}+\\dfrac 3{\\sqrt[3]{x}}\\right)^n$展开式中, 各项系数的和与各项二项式系数的和之差为$56$, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -99885,7 +100054,9 @@
     "003764": {
         "id": "003764",
         "content": "设常数$a\\in \\mathbf{R}$, 若$\\left(x^2+\\dfrac ax\\right)^5$的二项展开式中$x^7$的系数为$-10$, 则$a=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -101627,7 +101798,11 @@
     "003840": {
         "id": "003840",
         "content": "若$(1+x)^n+\\left(1+x^\\frac 12\\right)^n+\\left(1+x^\\frac 13\\right)^n+\\cdots+\\left(1+x^\\frac 1n\\right)^n \\ (n\\in \\mathbf{N}^*)$的展开式中$x$的系数是$a_n$, 展开式中所有项的系数和为$b_n$, 则$\\displaystyle\\lim_{n\\to \\infty}\\dfrac{na_n}{b_n}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X",
+            "K0820001X",
+            "K0820002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -101871,7 +102046,9 @@
     "003851": {
         "id": "003851",
         "content": "在$(1+x)^n$的展开式中, 第十项是使得二项式系数最大的唯一的项, 则$n$的值是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -102238,7 +102415,9 @@
     "003867": {
         "id": "003867",
         "content": "如果$\\left(\\dfrac{1}{\\sqrt{x}}-\\dfrac{\\sqrt{2}}{2}\\right)^n$展开式第三项的二项式系数为$66$, 那么展开式第六项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -102600,7 +102779,9 @@
     "003883": {
         "id": "003883",
         "content": "二项式$\\left(x+\\dfrac 1{2x}\\right)^8$展开式中的二项式系数最大的项的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -102979,7 +103160,9 @@
     "003899": {
         "id": "003899",
         "content": "设$A_n$为$(1+x)^{n+1}$的展开式中含$x^{n-1}$项的系数, $B_n$为$(1+x)^{n-1}$的展开式中二项式系数的和($n\\in \\mathbf{N}^*$), 则能使$A_n\\ge B_n$成立的$n$的最大值是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -103948,7 +104131,9 @@
     "003942": {
         "id": "003942",
         "content": "计算$1-3\\mathrm{C}_{10}^1+9\\mathrm{C}_{10}^2-27\\mathrm{C}_{10}^3+\\cdots-3^9\\mathrm{C}_{10}^9+3^{10}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -104426,7 +104611,9 @@
     "003962": {
         "id": "003962",
         "content": "设$(1+x)+(1+x)^2+\\cdots+(1+x)^n=a_0+a_1x+a_2x^2+\\cdots+a_nx^n, \\ n\\in \\mathbf{N}^*$, 若$a_1+a_2+\\cdots+a_{n-1}=61-n$, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -104737,7 +104924,9 @@
     "003976": {
         "id": "003976",
         "content": "设$\\left(3x^{\\frac 13}+x^\\frac 12\\right)^n$展开式的各项系数之和为$t$, 其二项式系数之和为$h$, 且$t+h=272$, 则展开式中$x^2$的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -105108,7 +105297,9 @@
     "003991": {
         "id": "003991",
         "content": "若$(x-2)^5=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$, 则$a_1+a_2+a_3+a_4+a_5=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -105251,7 +105442,9 @@
     "003997": {
         "id": "003997",
         "content": "如果$\\left(3x^2-\\dfrac{2}{x^3}\\right)^n$的展开式中含有非零常数项, 那么正整数$n$的最小值为\\bracket{20}.\n\\fourch{$10$}{$6$}{$5$}{$3$}",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -105789,7 +105982,8 @@
         "id": "004019",
         "content": "已知$(x^2+\\dfrac 1x)^n$的二项展开式的各项系数之和为$32$, 求该二项展开式中$x$的系数.",
         "objs": [
-            "K0819005X"
+            "K0819005X",
+            "K0820002X"
         ],
         "tags": [
             "第八单元",
@@ -106060,7 +106254,8 @@
         "id": "004030",
         "content": "利用二项式定理证明: 对于任意正整数$n$, $\\dfrac1{\\sqrt 5}[(2+\\sqrt 5)^n-(2-\\sqrt 5)^n]$都是正整数.",
         "objs": [
-            "K0819005X"
+            "K0819005X",
+            "K0819002X"
         ],
         "tags": [
             "第八单元",
@@ -108014,7 +108209,9 @@
     "004104": {
         "id": "004104",
         "content": "在$(x-\\sqrt 2)^8$的二项展开式中, $x^5$项的系数是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -108674,7 +108871,9 @@
     "004127": {
         "id": "004127",
         "content": "在$(x^2+\\dfrac 2x)^7$的二项展开式中, $x^2$的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -109239,7 +109438,9 @@
     "004148": {
         "id": "004148",
         "content": "若$(x+2)^n=x^n+ax^{n-1}+\\cdots +bx+c$($n\\in \\mathbf{N}^*$, $ n\\ge 3$), 且$b=4c$, 则$a$的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -109862,7 +110063,9 @@
     "004170": {
         "id": "004170",
         "content": "在二项式$(1+ax)^7(a\\in \\mathbf{R})$的展开式中, $x$的系数为$\\dfrac 73$, 则$\\displaystyle\\lim_{n\\to \\infty}(a+a^2+a^3+\\cdots +a^n)$的值是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -110429,7 +110632,9 @@
     "004192": {
         "id": "004192",
         "content": "已知$(x-\\dfrac 1{2x})^n$的展开式的常数项为第$6$项, 则常数项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -110896,7 +111101,9 @@
     "004211": {
         "id": "004211",
         "content": "在$(\\sqrt[3]x+\\dfrac 2x)^n$的二项展开式中, 所有项的系数之和为81, 则其常数项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -111438,7 +111645,9 @@
     "004231": {
         "id": "004231",
         "content": "二项式$(\\sqrt[3]x-\\dfrac 2x)^8$的展开式中的常数项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -111951,7 +112160,9 @@
     "004250": {
         "id": "004250",
         "content": "在$(x-\\dfrac 1{\\sqrt[3]x})^6$的二项展开式中, $x^2$项的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -113211,7 +113422,9 @@
     "004298": {
         "id": "004298",
         "content": "$(1+2x)^n$的二项展开式中, 含$x^3$项的系数等于含$x$项的系数的$8$倍, 则正整数$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -114388,7 +114601,8 @@
         "id": "004342",
         "content": "已知$a$是实数, 在$(1+ax)^8$的二项展开式中, 第$k+1$项的系数为$c_{k+1}=\\mathrm{C}_8^k\\cdot a^k \\ (k=0,1,2,3,\\cdots,8)$. 若$c_1<c_2<c_3<\\cdots<c_9$, 则$a$的取值范围为\\blank{50}.",
         "objs": [
-            "K0817006X"
+            "K0817006X",
+            "K0820003X"
         ],
         "tags": [
             "第四单元",
@@ -115775,7 +115989,9 @@
     "004394": {
         "id": "004394",
         "content": "已知$(x^2+\\dfrac 2x)^n$的二项展开式中, 所有二项式系数的和为$512$, 则展开式中的常数项为\\blank{50}(结果用数值表示).",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -116743,7 +116959,9 @@
     "004430": {
         "id": "004430",
         "content": "在$(x-a)^{10}$的展开式中, $x^7$的系数是$15$, 则实数$a=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -117291,7 +117509,9 @@
     "004450": {
         "id": "004450",
         "content": "在$(x+\\dfrac 1x)^{10}$的展开式中, 常数项等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -118118,7 +118338,9 @@
     "004475": {
         "id": "004475",
         "content": "设函数$f(x)=\\begin{cases} x^6, &  x\\ge 1,  \\\\ -2x-1, &  x\\le -1,  \\end{cases}$ 则当$x\\le -1$时, 则$f[f(x)]$表达式的展开式中含$x^2$项的系数是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -119360,7 +119582,9 @@
     "004517": {
         "id": "004517",
         "content": "在$(x+\\sqrt 2)^7$的二项展开式中任取一项, 则该项系数为有理数的概率为\\blank{50}(用数字作答).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -119837,7 +120061,9 @@
     "004536": {
         "id": "004536",
         "content": "已知$(1+ax)^6$的展开式中, 含有$x^3$项的系数为$160$, 则实数$a=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -120456,7 +120682,9 @@
     "004558": {
         "id": "004558",
         "content": "在$(x+\\dfrac{1}{\\sqrt{x}})^6$的二项展开式中, 常数项的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -122189,7 +122417,9 @@
     "004625": {
         "id": "004625",
         "content": "$(2+x)^6$的二项展开式中, 系数最大的项的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -123234,7 +123464,9 @@
     "004663": {
         "id": "004663",
         "content": "$(1+2x)^5=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$, 则$a_3=$\\blank{50}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -123619,7 +123851,10 @@
     "004678": {
         "id": "004678",
         "content": "已知代数式$(\\dfrac 2m+\\dfrac mx)^n$($m>0$, $x>0$).\\\\\n(1) 当$m=2$, $n=6$时, 求二项展开式中二项式系数最大的项;\\\\\n(2) 若$(\\dfrac 2m+\\dfrac mx)^{10}=a_0+\\dfrac{a_1}x+\\dfrac{a_2}{x^2}+\\cdots +\\dfrac{a_{10}}{x^{10}}$, 且$a_2=180$, 求$a_i$($0\\le i \\le 10$, $i\\in \\mathbf{N}$)的最大值.",
-        "objs": [],
+        "objs": [
+            "K0820003X",
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -123826,7 +124061,9 @@
     "004686": {
         "id": "004686",
         "content": "在$(1+2x)^6$的二项展开式中, $x^2$项的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -124481,7 +124718,10 @@
     "004711": {
         "id": "004711",
         "content": "若$(x+\\dfrac 1{\\sqrt{x}})^n$的二项展开式中各项系数的和等于$64$, 则其中$x^3$的系数是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820002X",
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -124890,7 +125130,9 @@
     "004727": {
         "id": "004727",
         "content": "$(1+2x)^{10}$ 的二项展开式中, $x^2$ 项的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -125391,7 +125633,9 @@
     "004747": {
         "id": "004747",
         "content": "在$(1+2x)^6$的二项展开式中, $x^5$项的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189597,7 +189841,9 @@
     "007526": {
         "id": "007526",
         "content": "求二项式$(2x-\\dfrac 3{2x^2})^7$展开式的第四项的二项式系数和笫四项的系数.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189619,7 +189865,9 @@
     "007527": {
         "id": "007527",
         "content": "求$(1+x)+(1+x)^2+(1+x)^3+\\cdots +(1+x)^{2n}$($n\\in \\mathbf{N}$)的展开式中念$x^n$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189643,7 +189891,9 @@
     "007528": {
         "id": "007528",
         "content": "在$(\\sqrt x+\\dfrac 1{\\sqrt[3]x})^{100}$的展开式中, 有多少项是有理项?",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189665,7 +189915,9 @@
     "007529": {
         "id": "007529",
         "content": "求$(x^2+\\dfrac 1{x^2}-2)^3$展开式中含$x^2$项的表达式.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189689,7 +189941,9 @@
     "007530": {
         "id": "007530",
         "content": "求$(1+x+x^2)(1-x)^{10}$展开式中含$x^4$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189713,7 +189967,9 @@
     "007531": {
         "id": "007531",
         "content": "求$(ax+by+cz)^n$的展开式中含$x^py^qz^r$项的系数, 其中$p+q+r=n$($p,q,r,n\\in \\mathbf{N}$).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189735,7 +189991,9 @@
     "007532": {
         "id": "007532",
         "content": "求$(x+\\dfrac 1x-1)^5$展开式中的常数项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189759,7 +190017,9 @@
     "007533": {
         "id": "007533",
         "content": "求证: $4^n-4^{n-1}\\mathrm{C}_n^1+4^{n-2}\\mathrm{C}_n^2-4^{n-3}\\mathrm{C}_n^3+\\cdots +4(-1)^{n-1}\\mathrm{C}_n^{n-1}+(-1)^n\\mathrm{C}_n^n=3^n$($n\\in \\mathbf{N}$).",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189781,7 +190041,9 @@
     "007534": {
         "id": "007534",
         "content": "求证: $1-\\mathrm{C}_n^2+\\mathrm{C}_n^4-\\mathrm{C}_n^6+\\mathrm{C}_n^8-\\mathrm{C}_n^{10}+\\cdots =(\\sqrt 2)^n\\cos \\dfrac{n\\pi }4$,\n$\\mathrm{C}_n^1-\\mathrm{C}_n^3+\\mathrm{C}_n^5-\\mathrm{C}_n^7+\\mathrm{C}_n^9-\\mathrm{C}_n^{11}+\\cdots =(\\sqrt 2)^n\\sin \\dfrac{n\\pi }4$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189803,7 +190065,9 @@
     "007535": {
         "id": "007535",
         "content": "求证: $\\mathrm{C}_n^1+2\\mathrm{C}_n^2+3\\mathrm{C}_n^3+\\cdots +n\\mathrm{C}_n^n=n\\cdot 2^{n-1}$($n\\in \\mathbf{N}$).",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189825,7 +190089,9 @@
     "007536": {
         "id": "007536",
         "content": "求证: $\\mathrm{C}_n^0+\\dfrac 12\\mathrm{C}_n^1+\\dfrac 13\\mathrm{C}_n^2+\\cdots +\\dfrac 1{n+1}\\mathrm{C}_n^n=\\dfrac 1{n+1}(2^{n+1}-1)$($n\\in \\mathbf{N}$).",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189847,7 +190113,9 @@
     "007537": {
         "id": "007537",
         "content": "求证$\\mathrm{C}_n^0\\mathrm{C}_n^1+\\mathrm{C}_n^1\\mathrm{C}_n^2+\\cdots +\\mathrm{C}_n^{n-1}\\mathrm{C}_n^n=\\dfrac{(2n)!}{(n-1)!(n+1)!}$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189871,7 +190139,9 @@
     "007538": {
         "id": "007538",
         "content": "求证: $(\\mathrm{C}_n^0)^2+(\\mathrm{C}_n^1)^2+(\\mathrm{C}_n^2)^2+\\cdots +(\\mathrm{C}_n^n)^2=\\mathrm{C}_{2n}^n$($n\\in \\mathbf{N}$).",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189893,7 +190163,9 @@
     "007539": {
         "id": "007539",
         "content": "求$53^{53}$除以$9$的余数.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189915,7 +190187,9 @@
     "007540": {
         "id": "007540",
         "content": "求证: $n^{n-1}-1$能被$(n-1)^2$整除($n\\ge 3$, $n\\in \\mathbf{N}$).",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189937,7 +190211,9 @@
     "007541": {
         "id": "007541",
         "content": "求证: $2<(1+\\dfrac 1n)^n<3$($n\\ge 2$, $n\\in \\mathbf{N}$).",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189959,7 +190235,9 @@
     "007542": {
         "id": "007542",
         "content": "在$(a-b)^n$($n\\in \\mathbf{N}$)的展开式中, 笫$r$项的二项式系数为\\bracket{20}.\n\\fourch{$\\mathrm{C}_n^r$}{$\\mathrm{C}_n^{r-1}$}{$(-1)^r\\mathrm{C}_n^r$}{$(-1)^{r-1}\\mathrm{C}_n^{r-1}$}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -189981,7 +190259,9 @@
     "007543": {
         "id": "007543",
         "content": "$(\\sqrt 3i-x)^{10}$展开式的第$8$项是\\bracket{20}.\n\\fourch{$-360\\sqrt 3x^7i$}{$-135x^3$}{$360\\sqrt 3x^7i$}{$3240\\sqrt 3x^3i$}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190003,7 +190283,9 @@
     "007544": {
         "id": "007544",
         "content": "$(\\dfrac 1{\\sqrt 3}-\\sqrt[3]x)^{20}$的展开式中, 不含$x$的项是\\bracket{20}.\n\\fourch{第$11$项}{第$12$项}{第$13$项}{第$7$项或第$13$项}",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190025,7 +190307,9 @@
     "007545": {
         "id": "007545",
         "content": "若二项式$(\\sqrt[3]x-\\dfrac 2x)^n$展开式中第$8$项是含$\\sqrt[3]x$的项, 则自然数$n$的值等于\\bracket{20}.\n\\fourch{$27$}{$28$}{$29$}{$30$}",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190047,7 +190331,9 @@
     "007546": {
         "id": "007546",
         "content": "在$(1+x)^n$的二项展开式中, 若第$9$项的系数与第$13$项的系数相等, 则第$20$项的系数等于\\bracket{20}.\n\\fourch{$19$}{$20$}{$21$}{$22$}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190069,7 +190355,9 @@
     "007547": {
         "id": "007547",
         "content": "若$(1+x)^8$展开式的中间三项依次成等差数列, 则$x$的值等于\\bracket{20}.\n\\fourch{$\\dfrac 12$或$2$}{$\\dfrac 12$或$4$}{$2$或$4$}{$2$或$\\dfrac 14$}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190091,7 +190379,9 @@
     "007548": {
         "id": "007548",
         "content": "在$(x-1)^9$按$x$降幂排列的展开式中, 系数最大的项是\\bracket{20}.\n\\fourch{第$4$项和第$5$项}{第$5$项}{第$5$项和第$6$项}{第$6$项}",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190113,7 +190403,10 @@
     "007549": {
         "id": "007549",
         "content": "在$(x+\\dfrac 2{x^2})^n$的展开式中, 第3项为常数, 则中间项的表达式为\\bracket{20}.\n\\fourch{$60$}{$160x^{-3}$}{$672$}{$960x^{-3}$}",
-        "objs": [],
+        "objs": [
+            "K0819002X",
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190135,7 +190428,9 @@
     "007550": {
         "id": "007550",
         "content": "$(x+1)^4-4(x+1)^3+6(x+1)^2-4(x+1)+1$等于\\bracket{20}.\n\\fourch{$x^4$}{$-x^4$}{$1$}{$-1$}",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190157,7 +190452,9 @@
     "007551": {
         "id": "007551",
         "content": "在$(x+y)^n$的展开式中, 若第$7$项的系数最大, 则$n$等于\\bracket{20}.\n\\fourch{$11, 12, 13$}{$13, 14$}{$11, 15$}{$12, 13$}",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190179,7 +190476,9 @@
     "007552": {
         "id": "007552",
         "content": "在$(x-\\dfrac 1x)^9$的展开式中, $x^3$的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190204,7 +190503,9 @@
     "007553": {
         "id": "007553",
         "content": "在$(ax+1)^7$的展开式中, 若$x^3$的系数是$x^2$的系数与$x^4$的系数的等差中项, 且$a>1$, 则$a$的值等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190228,7 +190529,9 @@
     "007554": {
         "id": "007554",
         "content": "在$(x+1+\\mathrm{i})^{10}$的展开式中, $x^6$的系数是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190250,7 +190553,9 @@
     "007555": {
         "id": "007555",
         "content": "若$a>0$, $n\\in \\mathbf{N}$, 且$(ax+1)^{2n}$和$(x+a)^{2n+1}$展开式的$x^n$的系数相等, 则$a$的収值范围是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190272,7 +190577,9 @@
     "007556": {
         "id": "007556",
         "content": "$(\\sqrt x+\\sqrt [3]{x^2})^{12}$的展开式的第5项是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190294,7 +190601,9 @@
     "007557": {
         "id": "007557",
         "content": "若二项式$(z-2)^6$展开式中的第$5$项是$-480$, 则复数$z$的值是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190316,7 +190625,9 @@
     "007558": {
         "id": "007558",
         "content": "若$(x+\\dfrac 1x)^n$展开式中的第$3$项和第$7$项系数相等, 则系数的最大项是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190338,7 +190649,9 @@
     "007559": {
         "id": "007559",
         "content": "在$(\\sqrt[3]a-\\dfrac 1{\\sqrt a})^{15}$的展开式中, 不含$a$的项是第\\blank{50}项.",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190360,7 +190673,9 @@
     "007560": {
         "id": "007560",
         "content": "$(\\dfrac{\\sqrt x}3+\\dfrac 3{\\sqrt x})^{12}$展开式的中间一项等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190382,7 +190697,9 @@
     "007561": {
         "id": "007561",
         "content": "$(2x^2+\\dfrac 1x)^{12}$展开式的常数项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190404,7 +190721,9 @@
     "007562": {
         "id": "007562",
         "content": "若$(\\dfrac 1{x\\sqrt[3]x}+x)^n$展开式中第$5, 6, 7$项的系数成等差数列, 则展开式中不含$x$的项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190426,7 +190745,9 @@
     "007563": {
         "id": "007563",
         "content": "在$(\\sqrt[3]2+\\sqrt 3)^{12}$的展开式中, 有理项是第\\blank{50}项.",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190448,7 +190769,9 @@
     "007564": {
         "id": "007564",
         "content": "在$(1-3x)^{12}$的展开式中, 各项的二项式系数之和为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190470,7 +190793,9 @@
     "007565": {
         "id": "007565",
         "content": "在$(1-x)^9$的展开式中, $x$的奇次项系数之和等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190495,7 +190820,9 @@
     "007566": {
         "id": "007566",
         "content": "若$(4x-1)^6=a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$, 则$a_6+a_5+a_4+a_3+a_2+a_1+a_0$的值等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190517,7 +190844,9 @@
     "007567": {
         "id": "007567",
         "content": "若$(1-2x)^6=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+a_6x^6$, 则$a_6-a_5+a_4-a_3+a_2-a_1$的值等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190539,7 +190868,9 @@
     "007568": {
         "id": "007568",
         "content": "在$(2x-1)^5$的展开式中, 各项系数的绝对值之和等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190565,7 +190896,9 @@
     "007569": {
         "id": "007569",
         "content": "在$(x+2y)(2x+y)^2(x+y)^3$的展开式中, 各项系数的和是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190587,7 +190920,9 @@
     "007570": {
         "id": "007570",
         "content": "$1+7\\mathrm{C}_n^1+7^2\\mathrm{C}_n^2+7^3\\mathrm{C}_n^3+\\cdots+7^n\\mathrm{C}_n^n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190609,7 +190944,9 @@
     "007571": {
         "id": "007571",
         "content": "$1-2\\mathrm{C}_n^1+4\\mathrm{C}_n^2-\\cdots +(-2)^n\\mathrm{C}_n^n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190631,7 +190968,9 @@
     "007572": {
         "id": "007572",
         "content": "$3+3^{n-1}\\mathrm{C}_n^1+3^{n-2}\\mathrm{C}_n^2+\\cdots +3\\mathrm{C}_n^{n-1}+\\mathrm{C}_n^n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190653,7 +190992,9 @@
     "007573": {
         "id": "007573",
         "content": "$\\mathrm{C}_{21}^0-\\mathrm{C}_{21}^2+\\mathrm{C}_{21}^4-\\mathrm{C}_{21}^6+\\cdots +\\mathrm{C}_{21}^{16}-\\mathrm{C}_{21}^{18}+\\mathrm{C}_{21}^{20}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190675,7 +191016,9 @@
     "007574": {
         "id": "007574",
         "content": "若$(2x^2-\\dfrac 1{\\sqrt[3]x})^n$的展开式中含有非零常数项, 则正整数$n$的最小值是\\bracket{20}.\n\\fourch{$8$}{$6$}{$5$}{$4$}",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190697,7 +191040,9 @@
     "007575": {
         "id": "007575",
         "content": "在$(\\sqrt[5]3+\\sqrt[7]5)^{24}$的展开式中, 整数项是\\bracket{20}.\n\\fourch{第$12$项}{第$13$项}{第$14$项}{第$15$项}",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190719,7 +191064,9 @@
     "007576": {
         "id": "007576",
         "content": "在$(\\sqrt 3x+\\sqrt[3]2)^{100}$的展开式中, $x$的系数为有理数的项共有\\bracket{20}.\n\\fourch{$15$项}{$16$项}{$17$项}{$18$项}",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190741,7 +191088,9 @@
     "007577": {
         "id": "007577",
         "content": "在$(1-x)^n(1+x)^n$的展开式中, 若含$x^4$项的系数是$10$, 则自然数$n$的值等于\\bracket{20}.\n\\fourch{$3$}{$4$}{$5$}{$6$}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190763,7 +191112,9 @@
     "007578": {
         "id": "007578",
         "content": "在二项式$(1+x)^n$的展开式中, 若相邻两项的系数之比为$8:15$, 则$n$的最小值是\\bracket{20}.\n\\fourch{$21$}{$22$}{$23$}{$24$}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190807,7 +191158,9 @@
     "007580": {
         "id": "007580",
         "content": "在$(2-3x)^n$的展开式中, 各项系数之和是\\bracket{20}.\n\\twoch{$1$}{$n$为偶数时是$2$, $n$为奇数时是$-2$}{$-1$}{$n$为偶数时是$1$, $n$为奇数时是$-1$}",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190829,7 +191182,9 @@
     "007581": {
         "id": "007581",
         "content": "在$(1+x)^3+(1+x)^4+\\cdots +(1+x)^{n+2}$的展开式中, 含$x^2$项的系数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_{n+3}^3$}{$\\mathrm{C}_{n+3}^3-1$}{$\\mathrm{C}_{n+2}^3-1$}{$\\mathrm{C}_{n+2}^3$}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190851,7 +191206,9 @@
     "007582": {
         "id": "007582",
         "content": "$(a+b+c)^{10}$展开式的项数共有\\bracket{20}.\n\\fourch{$11$项}{$66$项}{$121$项}{$132$项}",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190873,7 +191230,9 @@
     "007583": {
         "id": "007583",
         "content": "在$(x+1)(2x+1)(3x+1)\\cdots (nx+1)$的展开式中, $x$的一次项的系数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_n^1$}{$\\mathrm{C}_n^2$}{$\\mathrm{C}_{n+1}^1$}{$\\mathrm{C}_{n+1}^2$}",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190895,7 +191254,9 @@
     "007584": {
         "id": "007584",
         "content": "在$(1+x_1)(1+x_2)^2\\cdots (1+x_{n-1})^{n-1}(1+x_n)^n$展开式中, 各项系数之和是\\bracket{20}.\n\\fourch{$2^{n(n+1)}$}{$2^{\\frac{n(n+1)}2}$}{$2^{n+1}+2$}{$2(2^n-1)$}",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190917,7 +191278,9 @@
     "007585": {
         "id": "007585",
         "content": "$55^{55}$被$8$除所得的余数是\\bracket{20}.\n\\fourch{$7$}{$-7$}{$1$}{$-1$}",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190939,7 +191302,9 @@
     "007586": {
         "id": "007586",
         "content": "求$(x^2+\\dfrac 4{x^2}-4)^5$展开式中含$x^4$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190963,7 +191328,9 @@
     "007587": {
         "id": "007587",
         "content": "求$(x^2+3x+2)^5$展开式中含$x$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -190988,7 +191355,9 @@
     "007588": {
         "id": "007588",
         "content": "求$(1-x)^5(1+x+x^2)^4$展开式中含$x^7$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191010,7 +191379,9 @@
     "007589": {
         "id": "007589",
         "content": "求$(x-2)^4(1+x)^5$展开式中含$x^6$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191035,7 +191406,9 @@
     "007590": {
         "id": "007590",
         "content": "求$(x^2+x-2)^4$展开式中含$x^2$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191060,7 +191433,9 @@
     "007591": {
         "id": "007591",
         "content": "求$(2\\sqrt x-\\dfrac 1{\\sqrt x})^6$展开式中, $x$的一次幂的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191082,7 +191457,9 @@
     "007592": {
         "id": "007592",
         "content": "求$(x+y-3z)^8$的展开式中含$x^5yz^2$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191104,7 +191481,9 @@
     "007593": {
         "id": "007593",
         "content": "求$(x+2y+z)^9$展开式中含$x^2y^3z^4$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191128,7 +191507,9 @@
     "007594": {
         "id": "007594",
         "content": "求$(1-2x)^5(2+x)$展开式中含$x^3$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191150,7 +191531,9 @@
     "007595": {
         "id": "007595",
         "content": "求$(1+x+x^2)(1-x)^{10}$展开式中含$x^4$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191174,7 +191557,9 @@
     "007596": {
         "id": "007596",
         "content": "求$(1+x)^{2n}+x(1+x)^{2n-1}+x^2(1+x)^{2n-2}+\\cdots +x^n\\cdot (1+x)^n$展开式中含$x^n$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191196,7 +191581,9 @@
     "007597": {
         "id": "007597",
         "content": "求$(x-1)-(x-1)^2+(x-1)^3-(x-1)^4+(x-1)^5$的展开式中含$x^2$项的系数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191218,7 +191605,9 @@
     "007598": {
         "id": "007598",
         "content": "若$(x+x^{\\lg x})^5$的展开式的第$4$项为$10^6$, 求$x$的值.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191240,7 +191629,9 @@
     "007599": {
         "id": "007599",
         "content": "若$x(1-x)^4+x^2(1+2x)^k+x^3(1+3x)^{12}$的展开式中$x^4$的系数是$144$, 求$k$的值.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191262,7 +191653,9 @@
     "007600": {
         "id": "007600",
         "content": "若$(x^{\\lg x}+1)^n$展开式中最后$3$项的二项式系数的和是$22$, 而它的中间项是$20000$, 求$x$的值.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191284,7 +191677,9 @@
     "007601": {
         "id": "007601",
         "content": "已知$(x\\sin \\alpha +1)^6$的展开式中$x^2$项的系数与$(x-\\dfrac{15}2\\cos \\alpha)^4$的展开式中$x^3$项的系数相等, 求$\\alpha$的值.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191306,7 +191701,9 @@
     "007602": {
         "id": "007602",
         "content": "已知$(a+b)^n$展开式的末$3$项系数之和为$22$, 又$(x^{\\lg x}-3)^n$展开式的中间项等于$-540000$, 求$x$的值.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191328,7 +191725,9 @@
     "007603": {
         "id": "007603",
         "content": "求$(|x|+\\dfrac 1{|x|}-2)^3$展开式中的常数项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191352,7 +191751,9 @@
     "007604": {
         "id": "007604",
         "content": "求$[(1+\\log _3x)(1+\\log _x3)]^n$的展开式中不含$x$的项.",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191374,7 +191775,9 @@
     "007605": {
         "id": "007605",
         "content": "已知$(\\sqrt x+\\dfrac 2{x^2})^n$展开式中的第$5$项系数与第$3$项系数之比是$56:3$, 求展开式中不含$x$的项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191396,7 +191799,9 @@
     "007606": {
         "id": "007606",
         "content": "已知$(\\sqrt x+\\dfrac 1{2\\cdot \\sqrt[4]x})^n$展开式中前$3$项的系数依次成等差数列, 求展开式中所有的有理项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191418,7 +191823,9 @@
     "007607": {
         "id": "007607",
         "content": "已知$(x\\cdot \\sqrt x-\\dfrac 1x)^6$展开式的第$5$项等于$\\dfrac{15}2$, 求$\\displaystyle\\lim_{n\\to\\infty} (x^{-1}+x^{-2}+\\cdots +x^{-n})$.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191442,7 +191849,10 @@
     "007608": {
         "id": "007608",
         "content": "已知多项式$f(x)=(1+x)^m+(1+x)^n$($m\\in \\mathbf{N}$, $n\\in \\mathbf{N}$)的展开式中$x$项的系数为$19$.\\\\\n(1) 求$f(x)$中含$x^2$项的系数的最小值;\\\\\n(2) 对于使$f(x)$的$x^2$项的系数取最小值时的$m$, $n$, 求$f(x)$中含$x^7$的项.",
-        "objs": [],
+        "objs": [
+            "K0819005X",
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191464,7 +191874,9 @@
     "007609": {
         "id": "007609",
         "content": "在$(x+1)(x+2)(x+3)\\cdots (x+10)$的展开式中, $7$的系数是多少? $x^8$的系数又是多少?",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191486,7 +191898,9 @@
     "007610": {
         "id": "007610",
         "content": "求$(x+1)(x+2)(x+3)\\cdots (x+n)$展开式中含$x^{n-2}$项的系数.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191508,7 +191922,9 @@
     "007611": {
         "id": "007611",
         "content": "求多项式$(x^2+x-1)^9(2x+1)^4$展开式中$x$的奇次项系数之和.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191530,7 +191946,9 @@
     "007612": {
         "id": "007612",
         "content": "求多项式$(x^2+2x+2)^{1993}+(x^2-3x-3)^{1993}$展开式中$x$的偶次项系数之和.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191552,7 +191970,9 @@
     "007613": {
         "id": "007613",
         "content": "求$(2-5x+2x^2)^5(2-x)^7$展开后各项系数的和.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191574,7 +191994,9 @@
     "007614": {
         "id": "007614",
         "content": "求$(x^3+2x+1)(5x^2+4)$展开后各项系数的和.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191596,7 +192018,9 @@
     "007615": {
         "id": "007615",
         "content": "已知$(1+x)^n$展开式中奇数项之和为$A$, 偶数项之和为$B$, 试证: $A^2-B^2=(1-x^2)^n$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191618,7 +192042,9 @@
     "007616": {
         "id": "007616",
         "content": "若$(a+b)^n$展开式的所有奇数项的二项式系数之和为$1024$, 则展开式中间项的系数是\\bracket{20}.\n\\fourch{$330$}{$462$}{$682$}{$792$}",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191640,7 +192066,9 @@
     "007617": {
         "id": "007617",
         "content": "在$(x-\\dfrac 1x)^n$的展开式中, 若奇数项的系数之和为32, 则含$x^2$项的系数是\\bracket{20}.\n\\fourch{$-20$}{$-15$}{$15$}{$20$}",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191662,7 +192090,9 @@
     "007618": {
         "id": "007618",
         "content": "若$a$为常数, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{a+\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\cdots +\\mathrm{C}_n^n}{2^n}$的值等于\\bracket{20}.\n\\fourch{$0$}{$\\dfrac 12$}{$1$}{$\\dfrac a2$}",
-        "objs": [],
+        "objs": [
+            "K0820002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191684,7 +192114,9 @@
     "007619": {
         "id": "007619",
         "content": "记$(1+2x)^n$展开式中各项系数和为$a_n$, 其二项式系数和为$b_n$, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{b_n-a_n}{b_n+a_n}$为\\bracket{20}.\n\\fourch{$1$}{$0$}{$-1$}{不存在}",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191706,7 +192138,9 @@
     "007620": {
         "id": "007620",
         "content": "设$(1-2x)^8=a_0+a_1x+a_2x^2+\\cdots +a_8x^8$, 则$|a_0|+|a_1|+|a_2|+\\cdots +|a_8|$是\\bracket{20}.\n\\fourch{$-1$}{$1$}{$2^8$}{$3^8$}",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191728,7 +192162,9 @@
     "007621": {
         "id": "007621",
         "content": "在$(x-1)^{11}$的展开式中, $x$的偶次幂项的系数和为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191752,7 +192188,9 @@
     "007622": {
         "id": "007622",
         "content": "若$2000<\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\mathrm{C}_n^3+\\cdots +\\mathrm{C}_n^n<3000$, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191774,7 +192212,9 @@
     "007623": {
         "id": "007623",
         "content": "若$x^4-3x^3+x^2+1=a(x+1)^4+b(x+1)^3+c(x+1)^2+d(x+1)+6$, 则$b=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191819,7 +192259,9 @@
     "007625": {
         "id": "007625",
         "content": "设$(1+x)+(1+x)^2+(1+x)^3+\\cdots +(1+x)^n=b_0+b_1x+b_2x^2+\\cdots +b_nx^n$, 且$b_0+b_1+\\cdots +b_n=30$, 则自然数$n$的值等于\\bracket{20}.\n\\fourch{$4$}{$5$}{$6$}{$8$}",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191841,7 +192283,9 @@
     "007626": {
         "id": "007626",
         "content": "在$(x^2+x-1)^{100}+(x^2-x-1)^{100}$的展开式中, $x$的偶次项系数之和为\\bracket{20}.\n\\fourch{$4$}{$5$}{$6$}{$8$}",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191863,7 +192307,9 @@
     "007627": {
         "id": "007627",
         "content": "$\\mathrm{C}_n^0+2\\mathrm{C}_n^1+2^2\\mathrm{C}_n^2+\\cdots +2^n\\mathrm{C}_n^n$的值为\\bracket{20}.\n\\fourch{$2^n$}{$2^{n-1}$}{$3^n$}{$3^{n-1}$}",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191885,7 +192331,9 @@
     "007628": {
         "id": "007628",
         "content": "$101^{10}-1$的末尾连续零的个数是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191907,7 +192355,9 @@
     "007629": {
         "id": "007629",
         "content": "若$\\mathrm{C}_n^0(x+1)^n-\\mathrm{C}_n^1(x+1)^{n-1}+\\mathrm{C}_n^2(x+1)^{n-2}-\\cdots +(-1)^n\\mathrm{C}_n^n=a_0x^n+a_1x^{n-1}+\\cdots +a_{n-1}x+a_n$, 则$a_1+a_2+\\cdots +a_n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191929,7 +192379,9 @@
     "007630": {
         "id": "007630",
         "content": "已知$x$为实数, $i$为虚数单位, 则$(1+ix)^{50}$展开式中实系数项的系数和为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191951,7 +192403,9 @@
     "007631": {
         "id": "007631",
         "content": "设$a$是$\\sqrt 2$的整数部分, $b$是$\\sqrt 2$的小数部分, 则$(a-\\dfrac 1b)^6$展开式的中间项是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191973,7 +192427,9 @@
     "007632": {
         "id": "007632",
         "content": "设$(2x+x^{\\lg x})^n$展开式各项的二项式系数之和为$256$, 且二项式系数最大项的值为$1120$, 求$x$.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -191995,7 +192451,10 @@
     "007633": {
         "id": "007633",
         "content": "已知$(\\sqrt x+\\dfrac 1{\\sqrt[3]x})^n$展开式系数之和比$(a+b)^{2n}$展开式的系数之和小240, 求$(\\sqrt x+\\dfrac 1{\\sqrt[3]x})^n$展开式中系数最大的项.",
-        "objs": [],
+        "objs": [
+            "K0820001X",
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192061,7 +192520,9 @@
     "007636": {
         "id": "007636",
         "content": "求和: $\\mathrm{C}_{100}^0+4\\mathrm{C}_{100}^1+7\\mathrm{C}_{100}^2+\\cdots +(3n-2)\\mathrm{C}_{100}^{n-1}+\\cdots +298\\mathrm{C}_{100}^{99}+301\\mathrm{C}_{100}^{100}$($n\\in \\mathbf{N}$, $1\\le n\\le 101$).",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192083,7 +192544,9 @@
     "007637": {
         "id": "007637",
         "content": "设$a_0, a_1, a_2, \\cdots,a_n$是等差数列, 求证: $a_0+\\mathrm{C}_n^1a_1+\\mathrm{C}_n^2a_2+\\cdots +\\mathrm{C}_n^na_n=(a_0+a_n)\\cdot 2^{n-1}$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192105,7 +192568,9 @@
     "007638": {
         "id": "007638",
         "content": "若$n$为奇数, 求$7^n+\\mathrm{C}_n^1\\cdot 7^{n-1}+\\mathrm{C}_n^2\\cdot 7^{n-2}+\\mathrm{C}_n^37^{n-3}+\\cdots +\\mathrm{C}_n^{n-2}\\cdot 7^2+\\mathrm{C}_n^{n-1}\\cdot 7$被$9$除所得的余数.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192127,7 +192592,9 @@
     "007639": {
         "id": "007639",
         "content": "求$47^{13}$被$5$除的余数.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192149,7 +192616,9 @@
     "007640": {
         "id": "007640",
         "content": "求$91^{92}$除以$8$所得的余数.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192171,7 +192640,9 @@
     "007641": {
         "id": "007641",
         "content": "求证: $3^{2n}-8n-1$($n\\in \\mathbf{N}$)能被$64$整除.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192195,7 +192666,9 @@
     "007642": {
         "id": "007642",
         "content": "求证: 数列$65,65\\times 66, 65\\times 66^2, 65\\times 66^3, \\cdots, 65\\times 66^{48}, 65\\times 66^{49}$之和必能被$67$整除.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192217,7 +192690,9 @@
     "007643": {
         "id": "007643",
         "content": "已知$2^{n+2}\\times 3^n+5n-a$($n\\in \\mathbf{N}$)能被$25$整除, 求$a$的最小正值.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192239,7 +192714,9 @@
     "007644": {
         "id": "007644",
         "content": "求$x^{10}-3$除以$(x-1)^2$所得的余式.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192261,7 +192738,9 @@
     "007645": {
         "id": "007645",
         "content": "求证: 当$n\\ge 3$, $n\\in \\mathbf{N}$时, $n^{n-1}-1$能被$(n-1)^2$整除.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192283,7 +192762,9 @@
     "007646": {
         "id": "007646",
         "content": "设$(x-2)^8=a_8x^8+a_7x^7+\\cdots +a_1x+a_0$, 求$a_8+a_6+a_4+a_2$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192305,7 +192786,9 @@
     "007647": {
         "id": "007647",
         "content": "求$(1-x)+(1-x)^2+(1-x)^3+\\cdots +(1-x)^n$展开式中所有奇次项系数的和.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192327,7 +192810,9 @@
     "007648": {
         "id": "007648",
         "content": "已知$(3-x)^n=a_0+a_1x+a_2x^2+a_3x^3+\\cdots +a_nx^n$, 求$a_1+2a_2+2^2a_3+\\cdots +2^{n-1}a_n$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192349,7 +192834,9 @@
     "007649": {
         "id": "007649",
         "content": "求证: $\\mathrm{C}_n^0\\mathrm{C}_n^1+\\mathrm{C}_n^1\\mathrm{C}_n^2+\\cdots +\\mathrm{C}_n^{n-1}\\mathrm{C}_n^n=\\dfrac{(2n)!}{(n-1)!(n+1)!}$.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192373,7 +192860,9 @@
     "007650": {
         "id": "007650",
         "content": "求证: $\\mathrm{C}_n^0\\mathrm{C}_m^p+\\mathrm{C}_n^1\\mathrm{C}_m^{p-1}+\\cdots +\\mathrm{C}_n^p\\mathrm{C}_m^0=\\mathrm{C}_{m-n}^p$($p\\le m,n$).",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192395,7 +192884,9 @@
     "007651": {
         "id": "007651",
         "content": "利用$k\\mathrm{C}_n^k=n\\mathrm{C}_{n-1}^{k-1}$, 求证: $\\mathrm{C}_n^1+2\\mathrm{C}_n^2+3\\mathrm{C}_n^3+\\cdots +n\\mathrm{C}_n^n=n\\cdot 2^{n-1}$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192420,7 +192911,9 @@
     "007652": {
         "id": "007652",
         "content": "利用$k\\mathrm{C}_n^k=n\\mathrm{C}_{n-1}^{k-1}$, 求证: $\\mathrm{C}_n^1-2\\mathrm{C}_n^2+3\\mathrm{C}_n^3+\\cdots +(-1)^{n-1}n\\mathrm{C}_n^n=0$($n\\ge 2$, $n\\in \\mathbf{N}$).",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192444,7 +192937,9 @@
     "007653": {
         "id": "007653",
         "content": "利用$k\\mathrm{C}_n^k=n\\mathrm{C}_{n-1}^{k-1}$, 求证: $\\mathrm{C}_n^0+2\\mathrm{C}_n^1+3\\mathrm{C}_n^2+\\cdots +(n+1)\\mathrm{C}_n^n=(n+2)\\cdot 2^{n-1}$.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192468,7 +192963,9 @@
     "007654": {
         "id": "007654",
         "content": "已知$n\\in \\mathbf{N}$, $n\\ge 2$, 求证: $2^n>1+2+\\cdots +n$.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192490,7 +192987,9 @@
     "007655": {
         "id": "007655",
         "content": "求证: $3^n>2^{n-1}(n+2)$($n>2$, $n\\in \\mathbf{N}$).",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192512,7 +193011,9 @@
     "007656": {
         "id": "007656",
         "content": "已知正数$a,b,c$满足$a+b+c=abc$, 求证: $a^n+b^n+c^n>3(1+\\dfrac n2)$($n\\in \\mathbf{N}$).",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192534,7 +193035,9 @@
     "007657": {
         "id": "007657",
         "content": "利用数学归纳法证明: $(\\dfrac n2)^n>n!$($n\\in \\mathbf{N}$且$n\\ge 6$).",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192556,7 +193059,9 @@
     "007658": {
         "id": "007658",
         "content": "已知$\\mathrm{C}_{18}^n=\\mathrm{C}_{18}^{n+2}$, $4\\mathrm{P}_m^2=\\mathrm{P}_{m+1}^4$, 求$(1+\\sqrt m\\mathrm{i})^n$展开式中所有实数项的和.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192578,7 +193083,9 @@
     "007659": {
         "id": "007659",
         "content": "若实数$x,y$满足$x+y=1$, 求证: $x^5+y^5\\ge \\dfrac 1{16}$.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192600,7 +193107,9 @@
     "007660": {
         "id": "007660",
         "content": "已知: $|x|<1$, $n\\in \\mathbf{N}$, $n\\ge 2$, 求证: $(1-x)^n+(1+x)^n<2^n$.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192622,7 +193131,9 @@
     "007661": {
         "id": "007661",
         "content": "计算: $\\mathrm{C}_{21}^0-\\mathrm{C}_{21}^2+\\mathrm{C}_{21}^4-\\mathrm{C}_{21}^6+\\mathrm{C}_{21}^8-\\mathrm{C}_{21}^{10}+\\mathrm{C}_{21}^{12}-\\mathrm{C}_{21}^{14}+\\mathrm{C}_{21}^{16}-\\mathrm{C}_{21}^{18}+\\mathrm{C}_{21}^{20}$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192644,7 +193155,9 @@
     "007662": {
         "id": "007662",
         "content": "求证: $1+\\mathrm{C}_n^1\\cos \\alpha +\\mathrm{C}_n^2\\cos 2\\alpha +\\cdots +\\mathrm{C}_n^n\\cos n\\alpha =2^n\\cos ^n(\\dfrac{\\alpha }2)\\cdot \\cos \\dfrac{n\\alpha }2$,\n$\\mathrm{C}_n^1\\sin \\alpha +\\mathrm{C}_n^2\\sin 2\\alpha +\\cdots +\\mathrm{C}_n^n\\sin n\\alpha =2^n\\cos ^n(\\dfrac{\\alpha }2)\\sin \\dfrac{n\\alpha }2$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192666,7 +193179,9 @@
     "007663": {
         "id": "007663",
         "content": "设$a_n=1+q+q^2+\\cdots +q^{n-1}$($n\\in \\mathbf{N}$, $q\\ne \\pm 1$), $A_n=a_1\\mathrm{C}_n^1+a_2\\mathrm{C}_n^2+\\cdots +a_n\\mathrm{C}_n^n$.\\\\\n(1) 用$q,n$表示$A_n$;\\\\\n(2) 当$-3<q<1$时, 求$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{A_n}{2^n}$\\\\\n(3) 设$b_1+b_2+\\cdots +b_n=\\dfrac{A_n}{2^n}$, 求证: 数列$\\{b_n\\}$是等比数列.",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192688,7 +193203,9 @@
     "007664": {
         "id": "007664",
         "content": "设$A_n=(1+\\lg x)^n$, $B_n=1+n\\lg x+\\dfrac{n(n-1)}2\\lg ^2x$($n\\ge 3$, $n\\in \\mathbf{N}$), 且$x>\\dfrac 1{10}$, 试比较$A_n$和$B_n$的大小, 并证明你的结论.",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192953,7 +193470,9 @@
     "007676": {
         "id": "007676",
         "content": "设$(1+x+x^2)^n=a_0+a_1x+a_2x^2+\\cdots +a_{2n}x^{2n}$, 求$a_0+a_2+a_4+\\cdots +a_{2n}$的值.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192975,7 +193494,9 @@
     "007677": {
         "id": "007677",
         "content": "求$(\\sqrt x+2)^{2n+1}$的展开式中$x$的整数次幂的各项系数之和.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -192997,7 +193518,9 @@
     "007678": {
         "id": "007678",
         "content": "求$(1+i)^{4k-2}$($k\\in \\mathbf{N}$)展开式中奇数项之和.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -193019,7 +193542,9 @@
     "007679": {
         "id": "007679",
         "content": "求证: $(3+\\sqrt 7)^n$($n\\in \\mathbf{N}$, $n\\ge 2$)的整数部分为奇数.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231052,7 +231577,9 @@
     "009303": {
         "id": "009303",
         "content": "用二项式定理展开下列两式:\\\\\n(1) $(a+2b)^6$;\\\\\n(2) $(1-\\dfrac 1x)^5$.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231074,7 +231601,9 @@
     "009304": {
         "id": "009304",
         "content": "化简:\\\\\n(1) $(1+\\sqrt x)^5+(1-\\sqrt x)^5$;\\\\\n(2) $(2x+y)^4-(2x-y)^4$.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231096,7 +231625,9 @@
     "009305": {
         "id": "009305",
         "content": "(1) 求$(x-1)^{15}$的二项展开式中的前$4$项;\\\\\n(2) 求$(2a^3-3b^2)^{10}$的二项展开式中的第$8$项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231118,7 +231649,9 @@
     "009306": {
         "id": "009306",
         "content": "求下列各式的二项展开式中指定的项的系数:\\\\\n(1) $(1-\\dfrac 1{2x})^{10}$二项展开式中含$\\dfrac 1{x^4}$的项;\\\\\n(2) $(3x^3-\\dfrac 1{3x^3})^{10}$的二项展开式中的常数项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231140,7 +231673,9 @@
     "009307": {
         "id": "009307",
         "content": "在$(3x-2y)^9$的展开式中, 求二项式系数的和以及各项系数的和.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231162,7 +231697,9 @@
     "009308": {
         "id": "009308",
         "content": "(1) 用二项式定理证明: $(n+1)^n-1$能被$n^2$整除;\\\\\n(2) 用二项式定理证明: $99^{10}-1$能被$1000$整除.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231184,7 +231721,9 @@
     "009309": {
         "id": "009309",
         "content": "已知$(1+x)^n$的二项展开式中第$4$项与第$8$项的二项系数相等, 求这两项的二项式系数.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231206,7 +231745,9 @@
     "009310": {
         "id": "009310",
         "content": "求证: $2^n-\\mathrm{C}_n^1\\cdot 2^{n-1}+\\mathrm{C}_n^2\\cdot 2^{n-2}+\\cdots +\\mathrm{C}_n^{n-1}\\cdot 2+(-1)^n=1$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231228,7 +231769,9 @@
     "009311": {
         "id": "009311",
         "content": "$\\mathrm{C}_n^1+3\\mathrm{C}_n^2+9\\mathrm{C}_n^3+\\cdots +3^{n-1}\\mathrm{C}_n^n$等于\\bracket{20}.\n\\fourch{$4^n$}{$\\dfrac{4^n}3$}{$\\dfrac{4^n}3-1$}{$\\dfrac{4^n-1}3$}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231250,7 +231793,9 @@
     "009312": {
         "id": "009312",
         "content": "已知$n$为大于$1$的自然数, 证明: $(1+\\dfrac 1n)^n>2$.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231272,7 +231817,9 @@
     "009313": {
         "id": "009313",
         "content": "在$(x^2-\\dfrac 3x)^n$的二项展开式中, 有且只有第五项的二项式系数最大, 求$\\mathrm{C}_n^0-\\dfrac 12\\mathrm{C}_n^1+\\dfrac 14\\mathrm{C}_n^2-\\cdots +(-1)^n\\cdot \\dfrac 12\\mathrm{C}_n^n$.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231294,7 +231841,9 @@
     "009314": {
         "id": "009314",
         "content": "选择题:\n$\\mathrm{C}_{100}^0-\\mathrm{C}_{100}^2+\\mathrm{C}_{100}^4-\\cdots +\\mathrm{C}_{100}^{98}+\\mathrm{C}_{100}^{100}$等于\\bracket{20}.\n\\fourch{$-2^{50}$}{$0$}{$1$}{$2^{50}$}",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231316,7 +231865,9 @@
     "009315": {
         "id": "009315",
         "content": "求$(\\dfrac{\\sqrt x}2-\\dfrac 2{\\sqrt x})^{10}$的二项展开式的中间一项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231338,7 +231889,9 @@
     "009316": {
         "id": "009316",
         "content": "求$(x\\sqrt y-y\\sqrt x)^{11}$的二项展开式的中间两项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231360,7 +231913,9 @@
     "009317": {
         "id": "009317",
         "content": "在$(1+3x)^n$的二项展开式中, 末三项的二项式系数之和等于$631$.\\\\\n(1) 求二项展开式中二项式系数最大的项是第几项;\\\\\n(2) 求二项展开式中系数最大的项.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231382,7 +231937,9 @@
     "009318": {
         "id": "009318",
         "content": "求$77^{77}-15$除以$19$的余数.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231406,7 +231963,9 @@
     "009319": {
         "id": "009319",
         "content": "求证: $2^{6n-3}+3^{2n-1}$能被$11$整除.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231428,7 +231987,9 @@
     "009320": {
         "id": "009320",
         "content": "已知$(x+1)^n=x^n+\\cdots +ax^3+bx^2+cx+1$($n\\in \\mathbf{N}^*$), 且$a:b=3:1$, 求$c$的值.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231538,7 +232099,9 @@
     "009325": {
         "id": "009325",
         "content": "$(x^2-\\dfrac 1{2\\sqrt x})^3$的二项展开式的第$3$项是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231671,7 +232234,9 @@
     "009331": {
         "id": "009331",
         "content": "$(x-1)^n$的二项展开式中第$m$项($m\\le n$, $n\\in \\mathbf{N}^*$)的二项式的系数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_n^{m-1}$}{$(-1)^{m-1}\\mathrm{C}_n^m$}{$\\mathrm{C}_n^m$}{$(-1)^m\\mathrm{C}_n^m$}",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231737,7 +232302,9 @@
     "009334": {
         "id": "009334",
         "content": "已知$(x\\sin \\theta +1)^6$的二项展开式$x^2$项的系数与$(x-\\dfrac{15}2\\cos \\theta)^4$的二项展开式中$x^3$项的系数相等, 求$\\cos \\theta$的值.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231847,7 +232414,9 @@
     "009339": {
         "id": "009339",
         "content": "若$(1+\\sqrt x)^n$的展开式的系数和大于$8$且小于$32$, 则系数最大的项是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231935,7 +232504,9 @@
     "009343": {
         "id": "009343",
         "content": "求$\\mathrm{C}_{10}^1+2\\mathrm{C}_{10}^2+4\\mathrm{C}_{10}^3+\\cdots +2^9\\mathrm{C}_{10}^{10}$的值.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -231957,7 +232528,9 @@
     "009344": {
         "id": "009344",
         "content": "已知$(2^{\\lg x}-1)^n$的二项展开式中, 最后三项的二项式系数和等于$22$, 中间项为$-1280$, 求$x$的值.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -233367,7 +233940,9 @@
     "009407": {
         "id": "009407",
         "content": "已知$(x\\sqrt x-\\dfrac 1x)^6$的二项展开式的第$5$项为$\\dfrac{15}2$, 求$\\displaystyle\\lim_{n\\to\\infty}(x^{-1}+x^{-2}+\\cdots +x^{-n}) $的值.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -233391,7 +233966,9 @@
     "009408": {
         "id": "009408",
         "content": "设$n\\in \\mathbf{N}^*$, 求证: $\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\cdots +\\mathrm{C}_n^n=1+2+2^2+\\cdots +2^{n-1}$.",
-        "objs": [],
+        "objs": [
+            "K0820002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -233461,7 +234038,9 @@
     "009411": {
         "id": "009411",
         "content": "求$(1.009)^5$的近似值\\blank{50}(结果精确到$0.001$).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -233636,7 +234215,9 @@
     "009419": {
         "id": "009419",
         "content": "利用二项式定理证明: $3^n>2^{n-1}(n+2)$($n\\in \\mathbf{N}^*$, $n\\ge 2$).",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -233683,7 +234264,9 @@
     "009421": {
         "id": "009421",
         "content": "已知$(\\sqrt[3]x-\\dfrac 1{\\sqrt x})^n$的二项展开式中, 第三项与第二项的二项式系数之比为$11:2$ , 求正整数$n$及二项展开式中的所有的有理项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -233705,7 +234288,9 @@
     "009422": {
         "id": "009422",
         "content": "已知$(x^{\\lg x}+1)^n$的二项展开式中, 求三项的二项式系数的和为22. 二项式系数最大的项为20000, 求实数$x$的值.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -246011,7 +246596,9 @@
     "009945": {
         "id": "009945",
         "content": "(1) 求$(x-\\sqrt2y)^8$的二项展开式;\\\\\n(2) 求$(x-x^{-\\frac 13})^{12}$的二项展开式中的常数项;\\\\\n(3) 求$(x-\\dfrac 2x)^9$的二项展开式中$x^3$的系数;\\\\\n(4) 在$(1-x^2)^{20}$的二项展开式中, 如果第$4r$项和第$r+2$项的系数的绝对值相等, 求此展开式的第$4r$项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -246033,7 +246620,9 @@
     "009946": {
         "id": "009946",
         "content": "利用二项式定理证明: $7^{100}-1$是$8$的倍数.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -246055,7 +246644,9 @@
     "009947": {
         "id": "009947",
         "content": "(1) 若$(1-x)^6=a_0+a_1x+a_2x^2+\\cdots+a_6x^6$, 求$a_0+a_1+a_2+\\cdots+a_6$的值;\\\\\n(2) 已知$(x+1)^n=a_0+a_1(x-1)+a_2(x-1)^2+a_3(x-1)^3+\\cdots+a_n(x-1)^n$($n\\ge 2$, $n$为正整数), 求$a_0+a_1+a_2+\\cdots+a_n$的值.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -246077,7 +246668,9 @@
     "009948": {
         "id": "009948",
         "content": "(1) 求$(1+2x)^7$的二项展开式中系数最大的项;\\\\\n(2) 求$(1-2x)^7$的二项展开式中系数最大的项.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -246999,7 +247592,9 @@
     "009990": {
         "id": "009990",
         "content": "二项式$(3+x)^n$的展开式中, $x^2$项的系数是常数项的$5$倍, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819006X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -267096,7 +267691,7 @@
         "objs": [],
         "tags": [
             "第八单元",
-            "二项式定理"
+            "加法原理与乘法原理"
         ],
         "genre": "解答题",
         "ans": "",
@@ -267932,7 +268527,10 @@
     "010875": {
         "id": "010875",
         "content": "求$(2x^2-\\dfrac 1x)^6$的二项展开式中的中间项.",
-        "objs": [],
+        "objs": [
+            "K0819002X",
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -267954,7 +268552,9 @@
     "010876": {
         "id": "010876",
         "content": "求$(x+\\dfrac 1x)^{10}$的二项展开式中的常数项.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -267976,7 +268576,9 @@
     "010877": {
         "id": "010877",
         "content": "在$(\\sqrt x+\\dfrac 1{\\sqrt[3]x})^{24}$的二项展开式中, $x$的幂指数是负数的项一共有多少个?",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -267998,7 +268600,9 @@
     "010878": {
         "id": "010878",
         "content": "求$(x+\\dfrac 12)^8$的二项展开式中系数最大的项.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -268020,7 +268624,9 @@
     "010879": {
         "id": "010879",
         "content": "已知$x>0$, 且$(x+\\dfrac 1{x^3})^9$的二项展开式中, 第二项不大于第三项. 求实数$x$的取值范围.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -268042,7 +268648,9 @@
     "010880": {
         "id": "010880",
         "content": "已知$(1+x)^{10}=a_0+a_1(1-x)+a_2(1-x)^2+\\cdots+a_{10}(1-x)^{10}$, 求$a_8$的值.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -268064,7 +268672,9 @@
     "010881": {
         "id": "010881",
         "content": "求$(3-2x)^9$的二项展开式中系数最大的项.",
-        "objs": [],
+        "objs": [
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -268086,7 +268696,9 @@
     "010882": {
         "id": "010882",
         "content": "设$f(x)=(1+x)^m+(1+x)^n$($m$、$n$为正整数). 若二项展开式中关于$x$的一次项系数之和为$11$, 则当$m$、$n$为何值时, 含$x^2$项的系数取得最小值?",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -268108,7 +268720,9 @@
     "010883": {
         "id": "010883",
         "content": "在$(1+x)^n$的二项展开式中, 设奇数项之和为$A$, 偶数项之和为$B$. 求证: $A^2-B^2=(1-x^2)^n$.",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -270922,7 +271536,9 @@
     "010990": {
         "id": "010990",
         "content": "在$(1-2x)^6$的二项展开式中, $x^3$项的系数为\\blank{50}. (用数字作答)",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -271812,7 +272428,9 @@
     "011030": {
         "id": "011030",
         "content": "已知$(2x^2-\\dfrac 1x)^n$($n\\in \\mathbf{N}^*)$的展开式中各项的二项式系数之和为$128$, 则其展开式中含$\\dfrac 1x$项的系数是\\blank{50}.(结果用数值表示)",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -272382,7 +273000,9 @@
     "011054": {
         "id": "011054",
         "content": "已知二项式$(2x+\\dfrac 1x)^6$, 则其展开式中的常数项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -274428,7 +275048,9 @@
     "011137": {
         "id": "011137",
         "content": "二项式$(x-\\dfrac 1x)^6$的展开式中的常数项为\\blank{50}. (用数字作答)",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -276496,7 +277118,9 @@
     "011226": {
         "id": "011226",
         "content": "若$(x-\\dfrac ax)^9$的展开式中$x^3$的系数是$-84$, 则$a=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -277437,7 +278061,9 @@
     "011269": {
         "id": "011269",
         "content": "若$(ax^2+\\dfrac 1{\\sqrt x})^{5}$的展开式中的常数项为$-\\dfrac 5{2}$, 则实数$a$的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -277959,7 +278585,9 @@
     "011293": {
         "id": "011293",
         "content": "设$n\\in \\mathbf{N}^*$, 若$(2+\\sqrt x)^n$的二项展开式中, 有理项的系数之和为$365$, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -278287,7 +278915,9 @@
     "011308": {
         "id": "011308",
         "content": "已知二项式$(x^2+\\dfrac ax)^6$的展开式中含$x^3$项的系数是$160$, 则实数$a$的值是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -278989,7 +279619,9 @@
     "011340": {
         "id": "011340",
         "content": "在$(\\sqrt x+\\dfrac 2{x^2})^n$($n\\in \\mathbf{N}^*$)的展开式中, 设含有$(\\sqrt x)^{n-r}(\\dfrac 2{x^2})^r$的项为第$r+1$($0\\le r\\le n$, $n\\in \\mathbf{N}$)项. 若第$3$项与第$5$项的系数之比为$3:56$, 则展开式中的常数项为\\bracket{20}.\n\\fourch{$180$}{$160$}{$120$}{$100$}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -279137,7 +279769,9 @@
     "011347": {
         "id": "011347",
         "content": "$(x^2+\\dfrac 1x)^8$的展开式中$x^4$项的系数是\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -279610,7 +280244,9 @@
     "011369": {
         "id": "011369",
         "content": "$(x+\\dfrac 1x)^n$的展开式中的第$3$项为常数项, 则正整数$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819002X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -280242,7 +280878,9 @@
     "011396": {
         "id": "011396",
         "content": "$(1+\\dfrac 1{x^2})(1+x)^6$展开式中$x^2$项的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -280663,7 +281301,9 @@
     "011415": {
         "id": "011415",
         "content": "在$(1-2x)^{6}$的二项展开式中, $x^{3}$项的系数为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -281265,7 +281905,9 @@
     "011442": {
         "id": "011442",
         "content": "设$n\\in \\mathbf{N}^*$, $a_n$为$(x+2)^n-(x+1)^n$的展开式的各项系数之和, $m=-\\dfrac 12t+6$, , $b_n=[\\dfrac{a_1}3]+[\\dfrac{2a_2}{3^2}]+...+[\\dfrac{na_n}{3^n}]$($[x]$表示不超过实数$x$的最大整数), 则当$n$和$t$变动时, $(n-t)^2+(b_n-m)^2$的最小值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "第四单元",
@@ -281660,7 +282302,9 @@
     "011459": {
         "id": "011459",
         "content": "在$(x^2+\\dfrac 2x)^6$的二项展开式中, 常数项等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -282523,7 +283167,9 @@
     "011498": {
         "id": "011498",
         "content": "已知二项式$(2x+\\dfrac 1x)^6$, 则其展开式中的常数项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -283200,7 +283846,9 @@
     "011528": {
         "id": "011528",
         "content": "在$(\\dfrac 2x-x)^6$的二项展开式中, 常数项等于\\bracket{20}.\n\\fourch{$-160$}{$160$}{$-150$}{$150$}",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -285460,7 +286108,9 @@
     "011628": {
         "id": "011628",
         "content": "在$(x-\\dfrac 2x)^6$的二项展开式中, 常数项等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -285982,7 +286632,9 @@
     "011651": {
         "id": "011651",
         "content": "设常数$a\\in \\mathbf{R}$, 若$(x^2+\\dfrac ax)^5$的二项展开式中$x^7$项的系数为$-10$, 则$a=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -287117,7 +287769,9 @@
     "011703": {
         "id": "011703",
         "content": "在$(1+x+\\dfrac 1{x^{2015}})^{10}$的展开式中, $x^2$项的系数为\\blank{50}(结果用数值表示).",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -287546,7 +288200,9 @@
     "011723": {
         "id": "011723",
         "content": "在$(\\sqrt[3]x-\\dfrac 2x)^n$的二项式中, 所有项的二项式系数之和为$256$, 则常数项等于\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "KNONE"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -293357,7 +294013,9 @@
     "011993": {
         "id": "011993",
         "content": "$(1+x)^n$的二项展开式中, 若第9项与第13项系数相等, 则第20项为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -293809,7 +294467,9 @@
     "012011": {
         "id": "012011",
         "content": "$(1+2x)^5=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$, 则$a_3=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0819005X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -294232,7 +294892,10 @@
     "012026": {
         "id": "012026",
         "content": "已知代数式$(\\dfrac 2m+\\dfrac mx)^n$($m>0$,$x>0$).\\\\\n(1) 当$m=2$, $n=6$时, 求二项展开式中二项式系数最大的项;\\\\\n(2) 若$(\\dfrac 2m+\\dfrac mx)^{10}=a_0+\\dfrac{a_1}x+\\dfrac{a_2}{x^2}+\\cdots +\\dfrac{a_{10}}{x^{10}}$, 且$a_2=180$, 求$a_i$($0\\le i \\le 10$,$i\\in \\mathbf{N}$)的最大值.",
-        "objs": [],
+        "objs": [
+            "K0819005X",
+            "K0820003X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"
@@ -303578,7 +304241,9 @@
     "030071": {
         "id": "030071",
         "content": "已知$(a+3b)^n$的展开式中, 各项系数的和为$2^{n+6}$, 则$n=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0820001X"
+        ],
         "tags": [
             "第八单元",
             "二项式定理"