diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index 67dd88e0..46a579b5 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -113201,7 +113201,9 @@
             "20220701\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032063"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -352500,7 +352502,9 @@
             "20221215\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032075"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -357785,7 +357789,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032039"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -357986,7 +357992,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032042"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -358597,7 +358605,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032047"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -358669,7 +358679,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032048"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -359607,7 +359619,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032049"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -361608,7 +361622,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032069"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -365418,7 +365434,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032053"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -365483,7 +365501,7 @@
     },
     "013062": {
         "id": "013062",
-        "content": "将直线$l_1: n x+y-n=0$, $l_2: x+n y-n=0$($n \\in \\mathbf{N}$, $n \\geq 2$), $x$轴及$y$轴围成的封闭区域的面积记为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty} S_n=$\\blank{50}.",
+        "content": "将直线$l_1: n x+y-n=0$, $l_2: x+n y-n=0$($n \\in \\mathbf{N}$, $n \\geq 2$), $x$轴及$y$轴围成的封闭区域的面积记为$S_n$, 则$\\displaystyle\\lim_{n\\to+\\infty} S_n=$\\blank{50}.",
         "objs": [],
         "tags": [
             "第七单元",
@@ -365610,7 +365628,7 @@
     },
     "013066": {
         "id": "013066",
-        "content": "已知集合$M$是平面直角坐标系中方程为$x-2 k y+k^2=0$($k \\in \\mathbf{R}$)的直线的集合, 集合$S$是满足以下条件的点的集合: 对于集合$S$中的每一个点, 集合$M$中有且仅有一条直线经过该点.\\\\\n(1) 判断下列直线是否为集合$M$中的直线: $l_1: x-y+1=0$, $l_2: x-2 y+1=0$;\\\\\n(2) 判断下列各点是否为集合$S$中的点: $D(2,1)$, $E(1,1)$;\\\\\n(3) 求集合$S$中的点的轨迹方程.",
+        "content": "已知集合$M$是平面直角坐标系中方程为$x-2 k y+k^2=0$($k \\in \\mathbf{R}$)的直线的集合, 集合$S$是满足以下条件的点的集合: 对于集合$S$中的每一个点, 集合$M$中有且仅有一条直线经过该点.\\\\\n(1) 分别判断下列直线是否为集合$M$中的直线: $l_1: x-y+1=0$, $l_2: x-2 y+1=0$;\\\\\n(2) 分别判断下列各点是否为集合$S$中的点: $D(2,1)$, $E(1,1)$;\\\\\n(3) 求集合$S$中的点的轨迹方程.",
         "objs": [],
         "tags": [
             "第七单元",
@@ -366037,7 +366055,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032057"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -366187,7 +366207,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032060"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -366528,7 +366550,7 @@
     },
     "013096": {
         "id": "013096",
-        "content": "已知定圆$C_1:(x-7)^2+y^2=4$, $C_2:(x+7)^2+y^2=25$, 动圆$M$与两定圆外切, 则动圆圆心$M$的轨迹方程是\\blank{50}.",
+        "content": "已知定圆$C_1:(x-7)^2+y^2=4$, $C_2:(x+7)^2+y^2=25$, 动圆$M$与两定圆均外切, 则动圆圆心$M$的轨迹方程是\\blank{50}.",
         "objs": [],
         "tags": [
             "第七单元",
@@ -366564,7 +366586,7 @@
     },
     "013097": {
         "id": "013097",
-        "content": "若$F$是双曲线$x^2-y^2=1$的左焦点, 点$P$在第三象限的双曲线上, 则直线$FP$的倾斜角的取值范围是\\blank{50}.",
+        "content": "若$F$是双曲线$x^2-y^2=1$的左焦点, 点$P$在第三象限且在该双曲线上, 则直线$FP$的倾斜角的取值范围是\\blank{50}.",
         "objs": [],
         "tags": [
             "第七单元",
@@ -366636,7 +366658,7 @@
     },
     "013099": {
         "id": "013099",
-        "content": "记椭圆$E_n: \\dfrac{x^2}{4}+\\dfrac{n y^2}{4 n+1}=1$, 其中$n=1,2, \\cdots$. 当点$(x, y)$分别在$E_1, E_2, \\cdots$上时, $x+y$的最大值分别是$M_1, M_2, \\cdots$, 则$\\displaystyle\\lim_{n\\to\\infty} M_n=$\\blank{50}.",
+        "content": "记椭圆$E_n: \\dfrac{x^2}{4}+\\dfrac{n y^2}{4 n+1}=1$, 其中$n$为正整数. 当点$(x, y)$分别在$E_1, E_2, \\cdots$上时, $x+y$的最大值分别是$M_1, M_2, \\cdots$, 则$\\displaystyle\\lim_{n\\to +\\infty} M_n=$\\blank{50}.",
         "objs": [],
         "tags": [
             "第七单元",
@@ -367177,7 +367199,7 @@
     },
     "013118": {
         "id": "013118",
-        "content": "若直线$y=x+k$与曲线$y=\\sqrt{2-x^2}$相交于两点, 则实数$k$的取值范围为\\blank{50}.",
+        "content": "若直线$y=x+k$与曲线$y=\\sqrt{2-x^2}$恰有两个公共点, 则实数$k$的取值范围为\\blank{50}.",
         "objs": [],
         "tags": [
             "第七单元",
@@ -367377,7 +367399,7 @@
     },
     "013124": {
         "id": "013124",
-        "content": "已知直线$l: x-a y+a=0$与双曲线$x^2-y^2=1$的左支交于$A, B$两点, 过弦$AB$的中点$Q$与点$P(-2,1)$的直线交$y$轴于$(0, b)$点. 当$a$变化时, 求实数$b$的取值范围.",
+        "content": "已知直线$l: x-a y+a=0$与双曲线$x^2-y^2=1$的左支交于$A, B$两点, 过弦$AB$的中点$Q$与点$P(-2,1)$的直线$PQ$交$y$轴于$(0, b)$点. 当$a$变化时, 求实数$b$的取值范围.",
         "objs": [],
         "tags": [
             "第七单元",
@@ -367600,7 +367622,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032062"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -367788,7 +367812,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032058"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -368867,7 +368893,9 @@
             "20230118\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032050"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -383633,7 +383661,9 @@
             "20230128\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032045"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -383768,7 +383798,7 @@
     },
     "013758": {
         "id": "013758",
-        "content": "设$a, b$是两个实数, $A=\\{(x, y) | x=n, y=n a+b, n \\in \\mathbf{Z}\\}$, $B=\\{(x, y) | x=m, \\ y=3(m^2+5),\\ m \\in \\mathbf{Z}\\}$, $C=\\{x \\cdot y | x^2+y^2 \\leq 144\\}$, 讨论是否存在$a, b$使得$A \\cap B \\neq \\varnothing$且$(a, b) \\in C$.",
+        "content": "设$a, b$是两个实数, $A=\\{(x, y) | x=n, y=n a+b, n \\in \\mathbf{Z}\\}$, $B=\\{(x, y) | x=m, \\ y=3(m^2+5),\\ m \\in \\mathbf{Z}\\}$, $C=\\{(x, y) | x^2+y^2 \\leq 144\\}$, 讨论是否存在$a, b$使得$A \\cap B \\neq \\varnothing$且$(a, b) \\in C$.",
         "objs": [],
         "tags": [
             "第一单元",
@@ -388805,7 +388835,9 @@
             "20230128\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032070"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -389022,7 +389054,9 @@
             "20230128\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032054"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -389116,7 +389150,9 @@
             "20230128\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032055"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -391016,7 +391052,9 @@
             "20230128\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032052"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -392011,7 +392049,9 @@
             "20230128\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032076"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -392460,7 +392500,9 @@
             "20230128\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032077"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -393069,7 +393111,9 @@
             "20230131\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032040"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -393889,7 +393933,9 @@
             "20230131\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032043"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -393989,7 +394035,9 @@
             "20230131\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032044"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -394361,7 +394409,9 @@
             "20230131\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032046"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -399628,7 +399678,9 @@
             "20230202\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032056"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -401210,7 +401262,9 @@
             "20230209\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032051"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -402608,7 +402662,9 @@
             "20230209\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032064"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -402745,7 +402801,9 @@
             "20230209\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032065"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -404309,7 +404367,8 @@
         ],
         "same": [],
         "related": [
-            "023289"
+            "023289",
+            "032068"
         ],
         "remark": "",
         "space": "",
@@ -404404,7 +404463,9 @@
             "20230221\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032067"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -406001,7 +406062,9 @@
             "20230221\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032078"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -576217,7 +576280,9 @@
             "20230101\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032072"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -652895,7 +652960,9 @@
             "20221029\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032074"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -672627,7 +672694,8 @@
         ],
         "same": [],
         "related": [
-            "014108"
+            "014108",
+            "032041"
         ],
         "remark": "",
         "space": "",
@@ -672909,7 +672977,8 @@
         ],
         "same": [],
         "related": [
-            "013088"
+            "013088",
+            "032059"
         ],
         "remark": "",
         "space": "",
@@ -672947,7 +673016,8 @@
         ],
         "same": [],
         "related": [
-            "013100"
+            "013100",
+            "032061"
         ],
         "remark": "",
         "space": "",
@@ -673026,7 +673096,8 @@
         "same": [],
         "related": [
             "014003",
-            "031388"
+            "031388",
+            "032066"
         ],
         "remark": "",
         "space": "",
@@ -677246,7 +677317,8 @@
         ],
         "same": [],
         "related": [
-            "030409"
+            "030409",
+            "032073"
         ],
         "remark": "",
         "space": "",
@@ -692959,6 +693031,1076 @@
         "space": "4em",
         "unrelated": []
     },
+    "032039": {
+        "id": "032039",
+        "content": "已知集合$A=\\{y | y=x^2,\\ x \\in \\mathbf{R}\\}$, $B=\\{y | y=2^x,\\ x \\in \\mathbf{R}\\}$. 则$A \\cap B=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第二单元",
+            "第一单元",
+            "2023届高三-第二轮复习讲义-01-集合与逻辑"
+        ],
+        "genre": "",
+        "ans": "$\\{y | y>0\\}$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "012792"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032040": {
+        "id": "032040",
+        "content": "判断下列各题中甲是乙的什么条件, 并用相应序号填入空格内.(A.充分非必要条件、B.必要非充分条件、C.充要条件、D.既非充分又非必要条件).\\\\\n(1) 已知$\\triangle ABC$. 甲: $A>B$; 乙: $\\sin A>\\sin B$;\blank{50}\\\\\n(2) 已知$\\overrightarrow {a}$, $\\overrightarrow {b}$, $\\overrightarrow {c}$是非零的共面向量. 甲: $\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\overrightarrow {b} \\cdot \\overrightarrow {c}$; 乙: $\\overrightarrow {a}=\\overrightarrow {c}$;\blank{50}\\\\\n(3) 已知函数$y=f(x)$的定义域为$\\mathbf{R}$, 且存在导函数$f'(x)$. 甲: $f'(x)>0$恒成立; 乙: $y=f(x)$是$\\mathbf{R}$上的严格增函数.\blank{50}",
+        "objs": [],
+        "tags": [
+            "第一单元",
+            "2023届高三-第二轮复习讲义-01-集合与逻辑"
+        ],
+        "genre": "4em",
+        "ans": "(1) C; (2) B; (3) A",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课01-20240121修改",
+        "edit": [
+            "20230131\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014118"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032041": {
+        "id": "032041",
+        "content": "集合$A=\\{x | x^2-5 x-6=0\\}$, $B=\\{x | a x^2-x+6=0\\}$, 且$A \\cup B=A$. 则实数$a$的取值范围为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第一单元",
+            "2023届高三-第二轮复习讲义-01-集合与逻辑"
+        ],
+        "genre": "",
+        "ans": "$\\{0\\}\\cup (\\dfrac 1{24},+\\infty)$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课01-20230203修改-20240121修改",
+        "edit": [
+            "20230131\t王伟叶",
+            "20230203\t朱敏慧",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014108",
+            "031223"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032042": {
+        "id": "032042",
+        "content": "设常数$m \\in \\mathbf{R}$, 集合$A=\\{x | \\dfrac{6}{x+1} \\geq 1\\}$, $B=\\{x | x^2-2 x+2 m<0\\}$. 若$A \\cup B=A$, 求$m$的取值范围.",
+        "objs": [],
+        "tags": [
+            "第一单元",
+            "2023届高三-第二轮复习讲义-01-集合与逻辑"
+        ],
+        "genre": "4em",
+        "ans": "$[-\\dfrac 32,+\\infty)$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "012799"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032043": {
+        "id": "032043",
+        "content": "已知$f(x)=2 \\lg x-1, g(x)=2 \\lg x-3$.\\\\\n(1) 求不等式$f(x)<3$的解集;\\\\\n(2) 求$|f(x)|+|g(x)|$的最小值.",
+        "objs": [],
+        "tags": [
+            "第一单元",
+            "2023届高三-第二轮复习讲义-02-等式与不等式"
+        ],
+        "genre": "4em",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课03-20240121修改",
+        "edit": [
+            "20230131\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014153"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032044": {
+        "id": "032044",
+        "content": "若正数$a$、$b$满足$a b=1$, 则$\\dfrac{1}{2 a}+\\dfrac{1}{4 b}+\\dfrac{8}{a+2b}$的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第一单元",
+            "2023届高三-第二轮复习讲义-02-等式与不等式"
+        ],
+        "genre": "",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课03-20240121修改",
+        "edit": [
+            "20230131\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014157"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032045": {
+        "id": "032045",
+        "content": "已知$a, b\\in (0,+\\infty)$, 且$a \\neq b$, $n$是正整数. 求证: $(a+b)(a^n+b^n)<2(a^{n+1}+b^{n+1})$.",
+        "objs": [],
+        "tags": [
+            "第一单元",
+            "2023届高三-第二轮复习讲义-02-等式与不等式"
+        ],
+        "genre": "4em",
+        "ans": "证明略",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2020年空中课堂高三复习课03-20240121修改",
+        "edit": [
+            "20230128\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013752"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032046": {
+        "id": "032046",
+        "content": "用函数单调性的定义证明: 函数$y=\\log _2 \\dfrac{x-1}{x+1}$在区间$(1,+\\infty)$上是严格增函数.",
+        "objs": [],
+        "tags": [
+            "第二单元",
+            "2023届高三-第二轮复习讲义-03-幂指对函数"
+        ],
+        "genre": "4em",
+        "ans": "证明略",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课04-20240121修改",
+        "edit": [
+            "20230131\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014170"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032047": {
+        "id": "032047",
+        "content": "函数$d(x)=\\begin{cases}0, & x \\in \\mathbf{Q}, \\\\ 1, & x \\notin \\mathbf{Q},\\end{cases}$ 则$d(d(\\pi))=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第二单元",
+            "2023届高三-第二轮复习讲义-04-函数的概念与性质"
+        ],
+        "genre": "",
+        "ans": "$0$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "012824"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032048": {
+        "id": "032048",
+        "content": "已知函数$f(x)=\\begin{cases}x^2, & x \\geq 0,\\\\ -x^2, & x<0.\\end{cases}$ 若$f(2-a^2)>f(a)$, 则实数$a$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第二单元",
+            "2023届高三-第二轮复习讲义-04-函数的概念与性质"
+        ],
+        "genre": "",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "012826"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032049": {
+        "id": "032049",
+        "content": "设$a\\in \\mathbf{R}$,关于$x$的方程$x^2+2 x=(a+2) x+2$的两个实根为$x_1$、$x_2$, 记$M=|x_1-x_2|$.\\\\\n(1) 当$a\\in [-1,1]$时,求$M$的最大值;\\\\\n(2) 是否存在实数$m$, 使得不等式$m^2+t m+1 \\geq M$对任意$a \\in[-1,1]$及任意$t \\in[-1,1]$恒成立? 若存在, 求$m$的取值范围; 若不存在, 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第一单元",
+            "第二单元",
+            "2023届高三-四月错题重做-02-函数二",
+            "2023届高三-四月错题重做-02-易错题-函数2",
+            "2023届高三-第二轮复习讲义-04-函数的概念与性质"
+        ],
+        "genre": "4em",
+        "ans": "(1)暂缺;(2)$(-\\infty,-2]\\cup [2,+\\infty)$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "012856"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032050": {
+        "id": "032050",
+        "content": "给出下列命题, 其中正确命题的所有序号是\\blank{50}.\\\\\n\\textcircled{1} 直线上有两点到平面的距离相等, 则此直线与平面平行;\\\\\n\\textcircled{2} 夹在两个平行平面间的两条异面线段(端点均在相应平面上)的中点连线平行于这两个平面;\\\\\n\\textcircled{3} $\\alpha$内存在不共线的三点到$\\beta$的距离相等, 则平面$\\alpha$与$\\beta$平行;\\\\\n\\textcircled{4} 垂直于同一个平面的两条直线是平行直线.",
+        "objs": [],
+        "tags": [
+            "第六单元",
+            "2023届高三-第二轮复习讲义-09-立体几何综合"
+        ],
+        "genre": "",
+        "ans": "\\textcircled{2}\\textcircled{4}",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013178"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032051": {
+        "id": "032051",
+        "content": "如图, 在直径$AB=4$的半圆$O$内作一个内接直角三角形$ABC$, 使$\\angle BAC=30^{\\circ}$, 将图中阴影部分以直线$AB$为旋转轴旋转一周形成一个几何体, 则该几何体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [left] {$O$} coordinate (O) circle (0.03);\n\\draw (0,1.5) node [above] {$A$} coordinate (A);\n\\draw (0,-1.5) node [below] {$B$} coordinate (B);\n\\draw (-30:1.5) node [right] {$C$} coordinate (C);\n\\fill [pattern = north east lines] (A)--(C) arc (-30:90:1.5);\n\\fill [pattern = north east lines] (B)--(C) arc (-30:-90:1.5);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (A) arc (90:-90:1.5);\n\\draw pic [draw, \"$30^\\circ$\", angle eccentricity = 2] {angle = B--A--C};\n\\draw pic [draw, \"$60^\\circ$\", angle eccentricity = 1.5] {angle = C--B--A};\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元",
+            "2023届高三-第二轮复习讲义-09-立体几何综合"
+        ],
+        "genre": "",
+        "ans": "$\\dfrac{10}3\\pi$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课15-20240121修改",
+        "edit": [
+            "20230209\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014426"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032052": {
+        "id": "032052",
+        "content": "如图, 在四棱锥$P-ABCD$中, 已知$PA \\perp$平面$ABCD$, 且四边形$ABCD$为直角梯形, $\\angle ABC=\\angle BAD=\\dfrac{\\pi}{2}$, $PA=AD=2$, $AB=BC=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (1,0,0) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [left] {$Q$} coordinate (Q);\n\\draw (P)--(B) (P)--(C) (P)--(D) (B)--(C)--(D) (Q)--(C);\n\\draw [dashed] (B)--(A)--(D) (A)--(P); \n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$P-ABCD$的表面积;\\\\\n(2) 若$P, A, C, D$四点在同一球面上, 求该球的体积.",
+        "objs": [],
+        "tags": [
+            "第六单元",
+            "2023届高三-第二轮复习讲义-09-立体几何综合"
+        ],
+        "genre": "4em",
+        "ans": "(1) $\\dfrac 92+\\dfrac{\\sqrt{5}}2+\\sqrt{3}$; (2) $\\dfrac{8\\sqrt{2}}3\\pi$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2020年空中课堂高三复习课26-20240121修改",
+        "edit": [
+            "20230128\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014044"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032053": {
+        "id": "032053",
+        "content": "直线$2 x+3 y-1=0$的倾斜角是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-11-直线与圆"
+        ],
+        "genre": "",
+        "ans": "$\\pi-\\arctan\\dfrac {2}{3}}$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013059"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032054": {
+        "id": "032054",
+        "content": "若$k,-1, b$三个实数按此顺序构成等差数列, 则直线$y=k x+b$必经过定点\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元",
+            "第七单元",
+            "2023届高三-第二轮复习讲义-11-直线与圆"
+        ],
+        "genre": "",
+        "ans": "$(1,-2)$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2020年空中课堂高三复习课20-20240121修改",
+        "edit": [
+            "20230128\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013969"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032055": {
+        "id": "032055",
+        "content": "平面直角坐标系$xOy$中, 对于以点$(1,0)$为圆心且与直线$m x-y-2 m-1=0$($m \\in \\mathbf{R}$)相切的圆, 半径最大时圆的标准方程是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-11-直线与圆"
+        ],
+        "genre": "",
+        "ans": "$(x-1)^2+y^2=2$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2020年空中课堂高三复习课20-20240121修改",
+        "edit": [
+            "20230128\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013972"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032056": {
+        "id": "032056",
+        "content": "已知圆$N: (x-2)^2+(y+1)^2=4$.\\\\\n(1) 直线$l$经过点$A(3,2)$, 且被圆$N$截得长为$2 \\sqrt{2}$的弦, 求直线$l$的方程;\\\\\n(2) 过点$B(3,0)$的直线$l$与圆$N$交于$P$、$Q$两点, 直接写出弦$PQ$最短和最长时其所在直线的方程;\\\\\n(3) 讨论圆$C: (x-a)^2+y^2=1$($a \\in \\mathbf{R}$)与圆$N$的位置关系.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-11-直线与圆"
+        ],
+        "genre": "4em",
+        "ans": "(1) $x-y-1=0$或$7x+2y-23=0$; (2) 弦$PQ$最短时, 所在直线的方程为$x+y-3=0$; 弦$PQ$最长时, 所在直线的方程为$x-y+3=0$; (3) 当$a=2$时, 两圆内切; 当$a\\in (2-2\\sqrt{2},2)\\cup (2,2+2\\sqrt{2})$时, 两圆相交; 当$a=2\\pm 2\\sqrt{2}$时, 两圆外切; 当$a\\in (-\\infty,2-2\\sqrt{2})\\cup (2+2\\sqrt{2},+\\infty)$时, 两圆外离",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课18-20240121修改",
+        "edit": [
+            "20230202\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014366"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032057": {
+        "id": "032057",
+        "content": "设$m$是正实数, 若点$F(0,5)$是双曲线$\\dfrac{y^2}{m^2}-\\dfrac{x^2}{9}=1$的一个焦点, 则$m=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-12-圆锥曲线"
+        ],
+        "genre": "",
+        "ans": "$\\pm 4$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013079"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032058": {
+        "id": "032058",
+        "content": "已知点$M(x, y)$到点$F_1(-5,0)$和$F_2(5,0)$的距离差的绝对值是$8$, 则点$M$的轨迹方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-12-圆锥曲线"
+        ],
+        "genre": "",
+        "ans": "$\\dfrac{x^2}{16}-\\dfrac{y^2}{9}=1$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013137"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032059": {
+        "id": "032059",
+        "content": "过定点$F(4,0)$作直线$l$交$y$轴于$Q$点, 在$x$轴上存在一点$T$满足$\\overrightarrow{OT} \\perp \\overrightarrow{FQ}$, 延长$TQ$至$P$点, 使$\\overrightarrow{QT}=\\overrightarrow{PQ}$, 则动点$P$的轨迹方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-12-圆锥曲线"
+        ],
+        "genre": "",
+        "ans": "$y^2=16x",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20230214修改-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20230214\t吴惠群",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013088",
+            "031230"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032060": {
+        "id": "032060",
+        "content": "已知抛物线方程为$y^2=4x$, 过焦点$F$的直线与抛物线交于$A, B$两点, 以$AB$为直径的圆$M$与抛物线的准线$l$的位置关系为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-12-圆锥曲线"
+        ],
+        "genre": "",
+        "ans": "相切",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013083"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032061": {
+        "id": "032061",
+        "content": "抛物线$x^2=4 y$的焦点$F$, 过点$(0,-1)$作直线交抛物线于不同的两点$A, B$, 则弦$AB$的中点$M$的轨迹方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-12-圆锥曲线"
+        ],
+        "genre": "",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20230214修改-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20230214\t吴惠群",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013100",
+            "031231"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032062": {
+        "id": "032062",
+        "content": "若动点$(x, y)$在曲线$\\dfrac{x^2}{4}+y^2=1$上变化, 则$4 x+y^2$的最大值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-12-圆锥曲线"
+        ],
+        "genre": "",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013131"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032063": {
+        "id": "032063",
+        "content": "双曲线$C_1:\\dfrac{x^2}4-\\dfrac{y^2}{b^2}=1$与圆$C_2:x^2+y^2=4+b^2 \\ (b>0)$交于点$A(x_A,y_A)$(第一象限), 曲线$\\Gamma$由所有在$C_1$或$C_2$上, 且满足$|x|>x_A$的点组成, $C_2$与$x$轴的左、右交点分别记作$F_1,F_2$.\\\\\n(1) 若$x_A=\\sqrt6$, 求$b$的值;\\\\\n(2) 若$b=\\sqrt5$, 点$P$在曲线$\\Gamma$上, 且在第一象限, $|PF_1|=8$, 求$\\angle F_1PF_2$;\\\\\n(3) 点$D(0,\\dfrac{b^2}2+2)$, 过该点的直线斜率为$-\\dfrac b2$的$l$和$\\Gamma$有且只有两个交点, 求$b^2$的取值范围.",
+        "objs": [
+            "K0716002X",
+            "K0718001X"
+        ],
+        "tags": [
+            "第七单元",
+            "双曲线",
+            "2023届高三-第二轮复习讲义-12-圆锥曲线"
+        ],
+        "genre": "4em",
+        "ans": "(1) $2$; (2) $\\arccos\\dfrac{11}{16}$; (3) ",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "上海2020年秋季高考试题20-20240121修改",
+        "edit": [
+            "20220701\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "003629"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032064": {
+        "id": "032064",
+        "content": "若过点$O(0,0)$的直线$l$与抛物线$y^2=2 x$恰有一个公共点, 则直线$l$的方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-13-解析几何综合"
+        ],
+        "genre": "",
+        "ans": "$x=0$或$y=0$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课20-20240121修改",
+        "edit": [
+            "20230209\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014478"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032065": {
+        "id": "032065",
+        "content": "设椭圆$\\Gamma: \\dfrac{x^2}{a^2}+y^2=1$($a>0$), $F_1$、$F_2$分别是椭圆$\\Gamma$的左、右焦点, 椭圆$\\Gamma$的离心率为$\\dfrac{\\sqrt{2}}{2}$, 直线$l$与椭圆$\\Gamma$交于不同的两点$A$、$B$.\\\\\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 已知直线$l$经过椭圆$\\Gamma$的右焦点$F_2$, $P$、$Q$是椭圆$\\Gamma$上两点, 四边形$ABQP$是菱形, 求直线$l$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-第二轮复习讲义-13-解析几何综合"
+        ],
+        "genre": "4em",
+        "ans": "(1) $\\dfrac{x^2}2+y^2=1$; (2) $y=\\pm\\sqrt{2}(x-1)$; (3) $(\\dfrac{\\sqrt{6}}3,0)$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课20-20240121修改",
+        "edit": [
+            "20230209\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014483"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032066": {
+        "id": "032066",
+        "content": "设抛物线$C: y^2=4 x$的焦点为$F$, 过$F$且斜率为$k(k>0)$的直线$l$与$C$交于$A$、$B$两点, $|AB|=8$. 则直线$l$的方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元",
+            "2023届高三-四月错题重做-04-解析几何",
+            "2023届高三-第二轮复习讲义-13-解析几何综合"
+        ],
+        "genre": "",
+        "ans": "$y=x-1$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2020年空中课堂高三复习课22-20230214修改-20240121修改",
+        "edit": [
+            "20230128\t王伟叶",
+            "20230214\t吴惠群",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014003",
+            "031388",
+            "031233"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032067": {
+        "id": "032067",
+        "content": "某地区$2023$年产生的生活垃圾为$20$万吨, 其中$6$万吨垃圾以环保方式处理, 剩余$14$万吨垃圾以填埋方式处理. 预测显示: 在以$2023$年为第一年的未来十年内, 该地区每年产生的生活垃圾量比上一年增长$5 \\%$, 同时, 通过环保方式处理的垃圾量比上一年增加$1.5$万吨, 剩余的垃圾以填埋方式处理. 根据预测, 解答下列问题:\\\\\n(1) 求$2024$年至$2026$年, 该地区三年通过填埋方式处理的垃圾共计多少万吨? (结果精确到$0.1$万吨)\\\\\n(2) 该地区在哪一年通过环保方式处理的垃圾量首次超过这一年产生的生活垃圾量的$50 \\%$?",
+        "objs": [],
+        "tags": [
+            "第四单元",
+            "2023届高三-第二轮复习讲义-14-等差数列和等比数列"
+        ],
+        "genre": "4em",
+        "ans": "(1) 约$39.2$吨; (2) 在$2025$年, 该地区通过环保方式处理的垃圾量首次超过这一年产生的生活垃圾量的$50\\%$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课22-20240121修改",
+        "edit": [
+            "20230221\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014538"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032068": {
+        "id": "032068",
+        "content": "在等差数列$\\{a_n\\}$中, 公差为$2$, $11 a_5=5 a_8$, 则该数列前$n$项和$S_n$的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元",
+            "2023届高三-第二轮复习讲义-14-等差数列和等比数列"
+        ],
+        "genre": "",
+        "ans": "$-2d$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课22-20240121修改",
+        "edit": [
+            "20230221\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "023289",
+            "014535"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032069": {
+        "id": "032069",
+        "content": "数列$\\{a_n\\}$的通项公式$a_n=n \\cos \\dfrac{n \\pi}{2}+1$, 前$n$项和为$S_n$, 则$S_{2024}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元",
+            "2023届高三-第二轮复习讲义-15-数列综合"
+        ],
+        "genre": "",
+        "ans": "$3018$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届高三第二轮复习讲义-20240121修改",
+        "edit": [
+            "20230118\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "012923"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032070": {
+        "id": "032070",
+        "content": "已知数列$\\{a_n\\}$中, $a_1=3$, $a_2=5$, $\\{a_n\\}$的前$n$项和为$S_n$, 且满足$S_n+S_{n-2}=2S_{n-1}+2^{n-1}$($n \\geq 3$).\\\\\n(1) 试求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 令$b_n=\\dfrac{2^{n-1}}{a_n \\cdot a_{n+1}}$, $T_n$是数列$\\{b_n\\}$的前$n$项和, 证明: $T_n<\\dfrac{1}{6}$.",
+        "objs": [],
+        "tags": [
+            "第四单元",
+            "2023届高三-第二轮复习讲义-15-数列综合"
+        ],
+        "genre": "4em",
+        "ans": "(1) $a_n=2^n+1$; (2) 证明略",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2020年空中课堂高三复习课19-20240121修改",
+        "edit": [
+            "20230128\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "013960"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032071": {
+        "id": "032071",
+        "content": "设函数$y=f_i(x)$($i=1,2,3$)的导数分别为$y'=f_i'(x)$($i=1,2,3$).\\\\\n(1) 若$f_1(x)=\\sqrt{x}$, $f_2(x)=\\sqrt[3]{x}$, $f_3(x)=\\dfrac{1}{\\sqrt{x}}$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(2) 若$f_1(x)=x^3 \\cdot \\mathrm{e}^x$, $f_2(x)=x \\ln x$, $f_3(x)=\\dfrac{x}{\\sin x}$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(3) 若$f_1(x)=\\mathrm{e}^{2 x-1}$, $f_2(x)=2^x$, $f_3(x)=(\\dfrac{1}{3})^{x+1}$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(4) 若$f_1(x)=\\lg x$, $f_2(x)=\\ln (2 x-1)$, $f_3(x)=\\log _3(3-2 x)$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第二单元",
+            "2023届高三-第二轮复习讲义-16-导数及其应用"
+        ],
+        "genre": "",
+        "ans": "(1) $\\dfrac 12 x^{-\\frac 12}$; (2) $\\dfrac 13 x^{-\\frac 23}$; (3) $-\\dfrac 12 x^{-\\frac 32}$; (4) $\\mathrm{e}^x(x^3+3x^2)$; (5) $1+\\ln x$; (6) $\\dfrac{\\sin x-x\\cos x}{\\sin^2 x}$; (7) $2\\mathrm{e}^{2x-1}$; (8) $2^x\\ln 2$; (9) $-(\\dfrac 13)^{x+1}\\ln 3$; (10) $\\dfrac{1}{x\\ln 10}$; (11) $\\dfrac{2}{2x-1}$; (12) $\\dfrac{2}{(2x-3)\\ln 3}$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2024届高二下学期周末卷05-20240121修改",
+        "edit": [
+            "20230318\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "040202"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032072": {
+        "id": "032072",
+        "content": "求下列函数的最值.\\\\\n(1) $y=\\dfrac x{\\mathrm{e}^x}$, $x \\in[-1,+\\infty)$;\\\\\n(2) $y=x+2 \\cos x$, $x \\in[0, \\dfrac{\\pi}2]$.",
+        "objs": [],
+        "tags": [
+            "第二单元",
+            "导数",
+            "2023届高三-第二轮复习讲义-16-导数及其应用"
+        ],
+        "genre": "4em",
+        "ans": "(1) 最大值为$\\dfrac 1{\\mathrm{e}}$, 最小值为$-\\mathrm{e}$; (2)最大值为$\\dfrac{\\pi}6+\\sqrt{3}$, 最小值为$\\dfrac{\\pi}2$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2024届高二校本作业选择性必修第五章-20240121修改",
+        "edit": [
+            "20230101\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "021429"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032073": {
+        "id": "032073",
+        "content": "已知函数$f(x)=x^3-3ax^2+9x+5$($a<0$), 若曲线$y=f(x)$的的切线斜率最小时与直线$3x+y-2=0$平行, 则$a$的值为\\blank{50}.",
+        "objs": [
+            "K0232002X"
+        ],
+        "tags": [
+            "第二单元",
+            "导数",
+            "2023届高三-第二轮复习讲义-16-导数及其应用"
+        ],
+        "genre": "",
+        "ans": "$-2$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "高中数学教与学例题与习题-20230321修改-20240121修改",
+        "edit": [
+            "20221029\t王伟叶",
+            "20230321\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "030409",
+            "031356"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "032074": {
+        "id": "032074",
+        "content": "已知函数$f(x)=\\ln x-kx+1$, 若$f(x)\\le 0$恒成立, 求实数$k$的取值范围.",
+        "objs": [
+            "K0234003X"
+        ],
+        "tags": [
+            "第二单元",
+            "第四单元",
+            "导数",
+            "2023届高三-第二轮复习讲义-16-导数及其应用"
+        ],
+        "genre": "4em",
+        "ans": "$[1,+\\infty)$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "高中数学教与学例题与习题-20240121修改",
+        "edit": [
+            "20221029\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "030426"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032075": {
+        "id": "032075",
+        "content": "若函数$y=f(x)$($x \\in D$)同时满足下列两个条件, 则称$y=f(x)$在$D$上具有性质$M$.\\\\\n\\textcircled{1} $y=f(x)$在$D$上的导数$f'(x)$存在;\\\\\n\\textcircled{2} $y=f'(x)$在$D$上的导数$f''(x)$存在, 且$f''(x)>0$(其中$f''(x)=[f'(x)]'$)恒成立.\\\\\n 判断函数$y=\\lg \\dfrac 1x$在区间$(0,+\\infty)$上是否具有性质$M$? 并说明理由.",
+        "objs": [],
+        "tags": [
+            "第二单元",
+            "2023届高三-第二轮复习讲义-16-导数及其应用"
+        ],
+        "genre": "4em",
+        "ans": "具有性质$M$, 理由略",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023届普陀区一模试题21-20240121修改",
+        "edit": [
+            "20221215\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "012612"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032076": {
+        "id": "032076",
+        "content": "规定$\\mathrm{C}_x^m=\\dfrac{x(x-1) \\cdots(x-m+1)}{m !}$, 其中$x \\in \\mathbf{R}$, $m$是正整数, 且$\\mathrm{C}_x^0=1$, 这是组合数$\\mathrm{C}_n^m$($n$、$m$是正整数, 且$m \\leq n)$的一种推广， 则$\\mathrm{C}_{-15}^5$的值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第八单元",
+            "2023届高三-第二轮复习讲义-17-计数原理与二项式定理"
+        ],
+        "genre": "",
+        "ans": "$-11628$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2020年空中课堂高三复习课29-20240121修改",
+        "edit": [
+            "20230128\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014080"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032077": {
+        "id": "032077",
+        "content": "某人有$5$把钥匙, 其中有$1$把是房门钥匙, 但忘记了开房门的是哪一把. 于是, 他逐把不重复地试开, 问:\\\\\n(1) 恰好第三次打开房门锁的概率是多少?\\\\\n(2) 三次内打开的概率是多少?",
+        "objs": [],
+        "tags": [
+            "第八单元",
+            "2023届高三-第二轮复习讲义-18-概率与统计"
+        ],
+        "genre": "4em",
+        "ans": "(1) $\\dfrac 15$; (2) $\\dfrac 35$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2020年空中课堂高三复习课30-20240121修改",
+        "edit": [
+            "20230128\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014094"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "032078": {
+        "id": "032078",
+        "content": "实力相当的甲、乙两人参加乒乓球比赛, 规定$5$局$3$胜制(即$5$局内谁先赢$3$局就算胜出并停止比赛), 试求甲打完$5$局才获胜的概率.",
+        "objs": [],
+        "tags": [
+            "第八单元",
+            "2023届高三-第二轮复习讲义-18-概率与统计"
+        ],
+        "genre": "4em",
+        "ans": "(1) $\\dfrac 3{16}$",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2023年空中课堂高三复习课26-20240121修改",
+        "edit": [
+            "20230221\t王伟叶",
+            "20240121\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "014595"
+        ],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
     "040001": {
         "id": "040001",
         "content": "参数方程$\\begin{cases}x=3 t^2+4, \\\\ y=t^2-2\\end{cases}$($0 \\leq t \\leq 3$)所表示的曲线是\\bracket{20}.\n\\fourch{一支双曲线}{线段}{圆弧}{射线}",
@@ -698343,7 +699485,9 @@
             "20230318\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "032071"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []