收录高三寒假作业63新题

工具v2/文本文件/新题收录列表.txt

@@ -187,3 +187,6 @@
 20240125-141616
 024100:024107
 
+20240125-141926
+024108,020814,024109:024111,000315,024112
+

题库0.3/Problems.json

@@ -201195,7 +201195,9 @@
             "20220720\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "024109"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -388333,7 +388335,9 @@
             "20230128\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "024109"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -649030,6 +649034,109 @@
         "space": "4em",
         "unrelated": []
     },
+    "024108": {
+        "id": "024108",
+        "content": "若 $f(n)=1^2+2^2+\\cdots+n^2+(n+1)^2+n^2+\\cdots+2^2+1^2$, 则 $f(1)=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "024109": {
+        "id": "024109",
+        "content": "用数学归纳法证明``$1+\\dfrac{1}{2}+\\dfrac{1}{3}+\\cdots+\\dfrac{1}{2^n-1}<n$($n \\geq 2$)''时, 由 $n=k$ 的假设证明 $n=k+1$ 时, 不等式左边需增加的项数为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "006922",
+            "013931"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "024110": {
+        "id": "024110",
+        "content": "用数学归纳法证明 $\\dfrac{2^n-1}{2^n+1}>\\dfrac{n}{n+1}$ 对任意 $n \\geq k$($n, k \\in \\mathbf{N}$, $n$、$k \\geq 1$) 的自然数都成立, 则 $k$ 的最小值为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "024111": {
+        "id": "024111",
+        "content": "一个与正整数 $n$ 有关的命题, 当 $n=2$ 时命题成立, 且由 $n=k$($k \\geq 2$, $k \\in \\mathbf{N}$) 时命题成立可以推得 $n=k+2$ 时命题也成立, 则\\bracket{20}.\n\\twoch{该命题对于 $n>2$ 的自然数 $n$ 都成立}{该命题对于所有的正偶数都成立}{该命题何时成立与 $k$ 取值无关}{以上答案都不对}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "024112": {
+        "id": "024112",
+        "content": "用数学归纳法证明: $7^n+3^{n-1}$($n \\in \\mathbf{N}$, $n \\geq 1$) 能被 $4$ 整除.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
     "030001": {
         "id": "030001",
         "content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",