From c2e431d1b1f975d5e3da2c1bf3cf6afe8a1c751d Mon Sep 17 00:00:00 2001
From: "weiye.wang" <kj_wangweiye@126.com>
Date: Sat, 24 Jun 2023 20:46:00 +0800
Subject: [PATCH] =?UTF-8?q?=E6=94=B6=E5=BD=952023=E5=B1=8A=E5=8C=97?=
 =?UTF-8?q?=E4=BA=AC=E9=AB=98=E8=80=83=E8=AF=95=E5=8D=B7?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 工具/文本文件/批量题目分类号记录.txt |  14 +
 工具v2/批量收录题目.py               |   4 +-
 题库0.3/Problems.json                | 420 +++++++++++++++++++++++++++
 3 files changed, 436 insertions(+), 2 deletions(-)

diff --git a/工具/文本文件/批量题目分类号记录.txt b/工具/文本文件/批量题目分类号记录.txt
index eeb3b719..0be6ccc9 100644
--- a/工具/文本文件/批量题目分类号记录.txt
+++ b/工具/文本文件/批量题目分类号记录.txt
@@ -1,3 +1,17 @@
+20230624 2023届全国高考试卷
+
+problems_dict = {
+"2023届全国高考新高考I卷": "018039:018060",
+"2023年全国高考上海卷": "018061:018081",
+"2023届全国高考新高考II卷": "018082:018103",
+"2023届全国高考甲卷理科": "018104:018126",
+"2023届全国高考乙卷理科": "018127:018149",
+"2023届全国高考天津卷": "018150:018169",
+"2023届全国高考甲卷文科": "018170:018192",
+"2023届全国高考乙卷文科": "018193:018215",
+"2023届全国高考北京卷": "018237:018257"
+}
+
 20230621 2025届高一第一学期材料
 problems_dict = {
 "空中课堂必修第一册例题与习题": "011739:011987",
diff --git a/工具v2/批量收录题目.py b/工具v2/批量收录题目.py
index 8114b0e6..9f2e81c0 100644
--- a/工具v2/批量收录题目.py
+++ b/工具v2/批量收录题目.py
@@ -1,9 +1,9 @@
 #修改起始id,出处,文件名
 starting_id = 18237 #起始id设置, 来自"寻找空闲题号"功能
-raworigin = "测试一下" #题目来源的前缀(中缀在.tex文件中)
+raworigin = "" #题目来源的前缀(中缀在.tex文件中)
 filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目16.tex" #题目的来源.tex文件
 editor = "王伟叶" #编辑者姓名
-IndexDescription = " " #设置是否使用后缀, 留空("")则不用后缀, 不留空则以所设字符串作为后缀起始词, 按.tex文件中的顺序编号
+IndexDescription = "试题" #设置是否使用后缀, 留空("")则不用后缀, 不留空则以所设字符串作为后缀起始词, 按.tex文件中的顺序编号
 
 
 from database_tools import *
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index 87de915b..36f03ad3 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -467023,6 +467023,426 @@
         "space": "4em",
         "unrelated": []
     },
+    "018237": {
+        "id": "018237",
+        "content": "已知集合$M=\\{x | x+2 \\geq 0\\}$, $N=\\{x | x-1<0\\}$, 则$M \\cap N=$\\bracket{20}.\n\\fourch{$\\{x |-2 \\leq x<1\\}$}{$\\{x |-2<x \\leq 1\\}$}{$\\{x | x \\geq-2\\}$}{$\\{x | x<1\\}$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题1",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018238": {
+        "id": "018238",
+        "content": "在复平面内, 复数$z$对应的点的坐标是$(-1, \\sqrt{3})$, 则$z$的共轭复数$\\overline {z}=$\\bracket{20}.\n\\fourch{$1+\\sqrt{3} i$}{$1-\\sqrt{3} \\mathrm{i}$}{$-1+\\sqrt{3} i$}{$-1-\\sqrt{3} \\mathrm{i}$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题2",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018239": {
+        "id": "018239",
+        "content": "已知向量$\\overrightarrow {a}, \\overrightarrow {b}$满足$\\overrightarrow {a}+\\overrightarrow {b}=(2,3)$, $\\overrightarrow {a}-\\overrightarrow {b}=(-2,1)$, 则$|\\overrightarrow {a}|^2-|\\overrightarrow {b}|^2=$\\bracket{20}.\n\\fourch{$-2$}{$-1$}{$0$}{$1$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题3",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018240": {
+        "id": "018240",
+        "content": "下列函数中, 在区间$(0,+\\infty)$上单调递增的是\\bracket{20}.\n\\fourch{$f(x)=-\\ln x$}{$f(x)=\\dfrac{1}{2^x}$}{$f(x)=-\\dfrac{1}{x}$}{$f(x)=3^{|x-1|}$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题4",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018241": {
+        "id": "018241",
+        "content": "$(2 x-\\dfrac{1}{x})^5$的展开式中$x$的系数为\\bracket{20}.\n\\fourch{$-40$}{$40$}{$-80$}{$80$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题5",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018242": {
+        "id": "018242",
+        "content": "已知抛物线$C: y^2=8 x$的焦点为$F$, 点$M$在$C$上. 若$M$到直线$x=-3$的距离为$5$, 则$| MF |=$\\bracket{20}.\n\\fourch{$7$}{$6$}{$5$}{$4$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题6",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018243": {
+        "id": "018243",
+        "content": "在$\\triangle ABC$中, $(a+c)(\\sin A-\\sin C)=b(\\sin A-\\sin B)$, 则$\\angle C=$\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{6}$}{$\\dfrac{\\pi}{3}$}{$\\dfrac{2 \\pi}{3}$}{$\\dfrac{5 \\pi}{6}$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题7",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018244": {
+        "id": "018244",
+        "content": "若$x y \\neq 0$, 则``$x+y=0$''是``$\\dfrac{y}{x}+\\dfrac{x}{y}=-2$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题8",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018245": {
+        "id": "018245",
+        "content": "坡屋顶是我国传统建筑造型之一, 蕴含着丰富的数学元素. 安装灯带可以勾勒出建筑轮廓, 展现造型之美. 如图, 某坡屋顶可视为一个五面体, 其中两个面是全等的等腰梯形, 两个面是全等的等腰三角形. 若$AB=25 \\mathrm{m}$, $BC=10 \\mathrm{m}$, 且等腰梯形所在的平面、等腰三角形所在的平面与平面$ABCD$的夹角的正切值均为$\\dfrac{\\sqrt{14}}{5}$, 则该五面体的所有棱长之和为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.25]\n\\draw (-12.5,0,5) node [left] {$A$} coordinate (A);\n\\draw (12.5,0,5) node [right] {$B$} coordinate (B);\n\\draw (12.5,0,-5) node [right] {$C$} coordinate (C);\n\\draw (-12.5,0,-5) node [left] {$D$} coordinate (D);\n\\draw (A) ++ (5,{sqrt(14)},-5) node [above] {$F$} coordinate (F);\n\\draw (B) ++ (-5,{sqrt(14)},-5) node [above] {$E$} coordinate (E);\n\\draw (D)--(A)--(B)--(C)--(E)--(F)--cycle(A)--(F)(B)--(E);\n\\draw [dashed] (D)--(C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$102 \\mathrm{m}$}{$112 \\mathrm{m}$}{$117 \\mathrm{m}$}{$125 \\mathrm{m}$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题9",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018246": {
+        "id": "018246",
+        "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=\\dfrac{1}{4}(a_n-6)^3+6$($n=1,2,3, \\cdots$), 则\\bracket{20}.\n\\onech{当$a_1=3$时, $\\{a_n\\}$为递减数列, 且存在常数$M \\leq 0$, 使得$a_n>M$恒成立}{当$a_1=5$时, $\\{a_n\\}$为递增数列, 且存在常数$M \\leq 6$, 使得$a_n<M$恒成立}{当$a_1=7$时, $\\{a_n\\}$为递减数列, 且存在常数$M>6$, 使得$a_n>M$恒成立}{当$a_1=9$时, $\\{a_n\\}$为递增数列, 且存在常数$M>0$, 使得$a_n<M$恒成立}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题10",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018247": {
+        "id": "018247",
+        "content": "已知函数$f(x)=4^x+\\log _2 x$, 则$f(\\dfrac{1}{2})=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题11",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018248": {
+        "id": "018248",
+        "content": "已知双曲线$C$的焦点为$(-2,0)$和$(2,0)$, 离心率为$\\sqrt{2}$, 则$C$的方程为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题12",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018249": {
+        "id": "018249",
+        "content": "已知命题$p$: 若$\\alpha, \\beta$为第一象限角, 且$\\alpha>\\beta$, 则$\\tan \\alpha>\\tan \\beta$. 能说明$p$为假命题的一组$\\alpha, \\beta$的值为$\\alpha=$\\blank{50}, $\\beta=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题13",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018250": {
+        "id": "018250",
+        "content": "我国度量衡的发展有着悠久的历史, 战国时期就已经出现了类似于砝码的、用来测量物体质量的``环权''. 已知$9$枚环权的质量 (单位: 铢) 从小到大构成项数为$9$的数列$\\{a_n\\}$, 该数列的前$3$项成等差数列, 后$7$项成等比数列, 且$a_1=1$, $a_5=12$, $a_9=192$, 则$a_7=$\\blank{50}; 数列$\\{a_n\\}$所有项的和为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题14",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018251": {
+        "id": "018251",
+        "content": "设$a>0$, 函数$f(x)=\\begin{cases}x+2, & x<-a, \\\\ \\sqrt{a^2-x^2}, & -a \\leq x \\leq a,\\\\ -\\sqrt{x}-1, & x>a.\\end{cases}$ 给出下列四个结论:\\\\\n\\textcircled{1} $f(x)$在区间$(a-1,+\\infty)$上单调递减;\\\\\n\\textcircled{2} 当$a \\geq 1$时, $f(x)$存在最大值;\\\\\n\\textcircled{3} 设$M(x_1, f(x_1))$($x_1 \\leq a$), $N(x_2, f(x_2))$($x_2>a$), 则$|MN|>1$;\\\\\n\\textcircled{4} 设$P(x_3, f(x_3))$($x_3<-a$), $Q(x_4, f(x_4))$($x_4 \\geq -a$). 若$|PQ|$存在最小值, 则$a$的取值范围是$(0, \\dfrac{1}{2}]$.\\\\\n其中所有正确结论的序号是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题15",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018252": {
+        "id": "018252",
+        "content": "如图, 在三棱锥$P-ABC$中, $PA \\perp$平面$ABC$, $PA=AB=BC=1$, $PC=\\sqrt{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw ({sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw ({sqrt(2)/2},0,{sqrt(2)/2}) node [below] {$B$} coordinate (B);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp$平面$PAB$;\\\\\n(2) 求二面角$A-PC-B$的大小.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题16",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018253": {
+        "id": "018253",
+        "content": "设函数$f(x)=\\sin \\omega x \\cos \\varphi+\\cos \\omega x \\sin \\varphi$($\\omega>0$, $|\\varphi|<\\dfrac{\\pi}{2}$).\\\\\n(1) 若$f(0)=-\\dfrac{\\sqrt{3}}{2}$, 求$\\varphi$的值;\\\\\n(2) 已知$f(x)$在区间$[-\\dfrac{\\pi}{3}, \\dfrac{2 \\pi}{3}]$上单调递增, $f(\\dfrac{2 \\pi}{3})=1$, 再从条件\\textcircled{1}、条件\\textcircled{2}、条件\\textcircled{3}这三个条件中选择一个作为已知, 使函数$f(x)$存在, 求$\\omega, \\varphi$的值.\\\\\n条件\\textcircled{1}: $f(\\dfrac{\\pi}{3})=\\sqrt{2}$;\\\\\n条件\\textcircled{2}: $f(-\\dfrac{\\pi}{3})=-1$;\\\\\n条件\\textcircled{3}: $f(x)$在区间$[-\\dfrac{\\pi}{2},-\\dfrac{\\pi}{3}]$上单调递减.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题17",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018254": {
+        "id": "018254",
+        "content": "为研究某种农产品价格变化的规律, 收集得到了该农产品连续$40$天的价格变化数据, 如下表所示. 在描述价格变化时, 用``$+$''表示``上涨'', 即当天价格比前一天价格高; 用``$-$''表示``下跌'', 即当天价格比前一天价格低; 用``$0$''表示``不变'', 即当天价格与前一天价格相同. \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 时段  & \\multicolumn{10}{c|}{ 价格变化 } \\\\\n\\hline 第 1 天到第 10 天 &$-$&$+$&$+$&$0$&$-$&$-$&$-$&$+$&$+$&$0$ \\\\\n\\hline 第 11 天到第 20 天 & $+$&$0$&$-$&$-$&$+$&$-$&$+$&$0$&$0$&$+$\\\\\n\\hline 第 21 天到第 30 天 &$0$&$+$&$+$&$0$&$-$&$-$&$-$&$+$&$+$&$0$ \\\\\n\\hline 第31天到第40天 & $+$&$0$&$+$&$-$&$-$&$-$&$+$&$0$&$-$&$+$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n用频率估计概率.\\\\\n(1) 试估计该农产品价格``上涨''的概率;\\\\\n(2) 假设该农产品每天的价格变化是相互独立的. 在未来的日子里任取$4$天, 试估计该农产品价格在这$4$天中$2$天``上涨''、 $1$天``下跌''、 $1$天``不变''的概率;\\\\\n(3) 假设该农产品每天的价格变化只受前一天价格变化的影响. 判断第$41$天该农产品价格``上涨''``下跌''和``不变''的概率估计值哪个最大. (结论不要求证明)",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题18",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018255": {
+        "id": "018255",
+        "content": "已知椭圆$E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的离心率为$\\dfrac{\\sqrt{5}}{3}$, $A$、$C$分别是$E$的上、下顶点, $B$、$D$分别是$E$的左、右顶点, $|AC|=4$.\\\\\n(1) 求$E$的方程;\\\\\n(2) 设$P$为第一象限内$E$上的动点, 直线$PD$与直线$B C$交于点$M$, 直线$P A$与直线$y=-2$交于点$N$. 求证: $MN\\parallel CD$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题19",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018256": {
+        "id": "018256",
+        "content": "设函数$f(x)=x-x^3 \\mathrm{e}^{a x+b}$, 曲线$y=f(x)$在点$(1, f(1))$处的切线方程为$y=-x+1$.\\\\\n(1) 求$a, b$的值;\\\\\n(2) 设函数$g(x)=f'(x)$, 求$g(x)$的单调区间;\\\\\n(3) 求$f(x)$的极值点个数.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题20",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018257": {
+        "id": "018257",
+        "content": "已知数列$\\{a_n\\}, \\{b_n\\}$的项数均为$m$($m>2$), 且$a_n, b_n \\in\\{1, 2, \\cdots, m\\}$, $\\{a_n\\}, \\{b_n\\}$的前$n$项和分别为$A_n, B_n$, 并规定$A_0=B_0=0$. 对于$k \\in\\{0, 1, 2, \\cdots, m\\}$, 定义$r_k=\\max \\{i | B_i \\leq A_k,\\ i \\in\\{0, 1, 2, \\cdots, m\\}\\}$, 其中, $\\max M$表示数集$M$中最大的数.\\\\\n(1) 若$a_1=2$, $a_2=1$, $a_3=3$, $b_1=1$, $b_2=3$, $b_3=3$, 求$r_0, r_1, r_2, r_3$的值;\\\\\n(2) 若$a_1 \\geq b_1$, 且$2 r_j \\leq r_{j+1}+r_{j-1}$, $j=1, 2, \\cdots, m-1$, 求$r_n$;\\\\\n(3) 证明: 存在$p, q, s, t \\in\\{0, 1, 2, \\cdots, m\\}$, 满足$p>q$, $s>t$, 使得$A_p+B_t=A_q+B_s$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届全国高考北京卷试题21",
+        "edit": [
+            "20230624\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
     "020001": {
         "id": "020001",
         "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",