diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index f0b30ea5..f573c883 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -632164,7 +632164,9 @@
         "id": "023556",
         "content": "$3$和$7$的等差中项是\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632184,7 +632186,9 @@
         "id": "023557",
         "content": "陈述句``$a=0$ 且 $b=0$''的否定形式为\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第一单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632204,7 +632208,9 @@
         "id": "023558",
         "content": "数列$\\{a_n\\}$是等差数列, $a_1=1$,公差$d=2$, 该数列的前$10$项和$S_{10}=$\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632224,7 +632230,9 @@
         "id": "023559",
         "content": "已知$\\log_2 5=a$, 则$\\log_2 25=$\\blank{50}(请用$a$表示).",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第二单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632244,7 +632252,9 @@
         "id": "023560",
         "content": "函数$f(x)=2^x+m$的反函数为$y=f^{-1}(x)$, 且$y=f^{-1}(x)$的图像过点$Q(5,2)$, 那么实数$m=$\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第二单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632264,7 +632274,9 @@
         "id": "023561",
         "content": "函数$y=\\sqrt{-2x^2+3x-1}$的定义域是\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第二单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632284,7 +632296,9 @@
         "id": "023562",
         "content": "无穷等比数列首项为$2$,公比为$q \\ (0<q<1)$, 前$n$项和为$S_n$, 若$\\displaystyle\\lim_{n\\to+\\infty}S_n=3$, 则$q=$\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632304,7 +632318,9 @@
         "id": "023563",
         "content": "数列$\\{a_n\\}$前$n$项和为$2^n-2$, 则通项$a_n=$\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632324,7 +632340,9 @@
         "id": "023564",
         "content": "等差数列$\\{a_n\\}$中, $a_{20}<0$, $a_{20}+a_{21}>0$. 设$S_n$是数列$\\{a_n\\}$的前$n$项和, 若$S_k>0$, 则正整数$k$的最小值为\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632342,9 +632360,12 @@
     },
     "023565": {
         "id": "023565",
-        "content": "设常数$m\\in \\mathbf{R}$.关于$x$的方程$\\sqrt{2x}=x+m$有两个不同的实数解, 则$m$的取值范围是\\blank{50}.",
+        "content": "设常数$m\\in \\mathbf{R}$. 关于$x$的方程$\\sqrt{2x}=x+m$有两个不同的实数解, 则$m$的取值范围是\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第二单元",
+            "第一单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632362,9 +632383,11 @@
     },
     "023566": {
         "id": "023566",
-        "content": "已知函数$f(x)$是定义在$\\mathbf{R}$上的严格增函数,且$y=f(x)$是奇函数.  若关于$x$的不等式$f(m x)+f(-x^2-2)<0$在区间$[1,5]$上恒成立, 则实数$m$的取值范围为\\blank{50}.",
+        "content": "已知函数$f(x)$是定义在$\\mathbf{R}$上的严格增函数, 且$y=f(x)$是奇函数.  若关于$x$的不等式$f(m x)+f(-x^2-2)<0$在区间$[1,5]$上恒成立, 则实数$m$的取值范围为\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第二单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632384,7 +632407,9 @@
         "id": "023567",
         "content": "已知数列 $\\{a_n\\}$ 的各项均为正数, 其前 $n$ 项和 $S_n$ 满足 $a_n \\cdot S_n=9$($n=1,2, \\cdots$). 给出下列四个结论:\n\\textcircled{1} $\\{a_n\\}$ 的第 2 项小于 3 ;\n\\textcircled{2} $\\{a_n\\}$ 为等比数列;\n\\textcircled{3} $\\{a_n\\}$ 为严格减数列;\n\\textcircled{4} $\\{a_n\\}$ 中存在小于 $\\dfrac{1}{100}$ 的项.其中所有正确结论的序号是\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632404,7 +632429,9 @@
         "id": "023568",
         "content": "已知实数$a,b$满足$a>b$, 则下列不等式中恒成立的是\\bracket{20}.\n\\fourch{$a^2>b^2$}{$\\dfrac 1a<\\dfrac 1b$}{$|a|>|b|$}{$2^a>2^b$}",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第一单元"
+        ],
         "genre": "选择题",
         "ans": "",
         "solution": "",
@@ -632424,7 +632451,9 @@
         "id": "023569",
         "content": "``$a=1$''是``函数$f(x)=|x-a|$在区间$[1,+\\infty)$上为严格增函数''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既非充分又非必要条件}",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第二单元"
+        ],
         "genre": "选择题",
         "ans": "",
         "solution": "",
@@ -632444,7 +632473,9 @@
         "id": "023570",
         "content": "斐波那契数列$\\{a_n\\}$满足$a_1=a_2=1$, $a_{n+2}=a_{n+1}+a_n(n\\geq 1, n\\in \\mathbf{N})$, 设$a_1+a_3+a_5+a_7+a_9+\\cdots+a_{2023}=a_k$, 则$k=$\\bracket{20}.\n\\twoch{2022}{2023}{2024}{2025}",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "选择题",
         "ans": "",
         "solution": "",
@@ -632464,7 +632495,9 @@
         "id": "023571",
         "content": "定义域和值域均为$[-a, a]$(常数$a>0$) 的函数$y=f(x)$和$y=g(x)$的图像如图所示, 给出下列四个命题: \\textcircled{1} 方程$f(g(x))=0$有且仅有三个解; \\textcircled{2} 方程$g(f(x))=0$有且仅有三个解; \\textcircled{3} 方程$f(f(x))=0$有且仅有九个解; \\textcircled{4} 方程$g(g(x))=0$有且仅有一个解. 那么, 其中正确命题的序号为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2) rectangle (2,2);\n\\draw (-2,-2) .. controls +(75:1) and +(180:0.6) .. (-1,0.6) .. controls +(0:0.6) and +(225:1.5) .. (1,0) .. controls +(45:0.5) and +(255:1) .. (2,2);\n\\filldraw (1,0) circle (0.03);\n\\draw (1,0) node [below] {\\tiny $\\dfrac a2$};\n\\draw (0.1,1) -- (0,1) node [right] {\\tiny $\\dfrac a2$};\n\\draw (-1,1) node [above] {\\small $y=f(x)$};\n\\draw (-2,0) node [below left] {$-a$};\n\\draw (2,0) node [below right] {$a$};\n\\draw (0,2) node [above right] {$a$};\n\\draw (0,-2) node [below right] {$-a$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2) rectangle (2,2);\n\\draw (-2,2) .. controls +(-45:1) and +(165:0.6) .. (0,0.5) .. controls +(-15:0.6) and +(135:0.5) .. (1,0) .. controls +(-45:0.5) and +(105:1) .. (2,-2);\n\\filldraw (1,0) circle (0.03);\n\\draw (1,0) node [below] {\\tiny $\\dfrac a2$};\n\\draw (0.1,1) -- (0,1) node [left] {\\tiny $\\dfrac a2$};\n\\draw (1,1) node [above] {\\small $y=g(x)$};\n\\draw (-2,0) node [below left] {$-a$};\n\\draw (2,0) node [below right] {$a$};\n\\draw (0,2) node [above right] {$a$};\n\\draw (0,-2) node [below right] {$-a$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{ \\textcircled{1}  \\textcircled{3} }{ \\textcircled{1}  \\textcircled{4} }{ \\textcircled{2}  \\textcircled{3} }{ \\textcircled{2}  \\textcircled{4} }",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第二单元"
+        ],
         "genre": "选择题",
         "ans": "",
         "solution": "",
@@ -632484,7 +632517,9 @@
         "id": "023572",
         "content": "已知函数$f(x)=x^2-\\dfrac{1}{x}$.\\\\\n(1) 判断函数$f(x)$是否是偶函数，并说明理由;\\\\\n(2) 判断$f(x)$在$(0, +\\infty)$上的单调性，并说明理由.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第二单元"
+        ],
         "genre": "解答题",
         "ans": "",
         "solution": "",
@@ -632504,7 +632539,9 @@
         "id": "023573",
         "content": "已知数列$\\{a_n\\}$的各项均不为零, 且$a_{n+1}=\\dfrac{3a_n}{a_n+3}$, $b_n=\\dfrac{1}{a_n}$. \\\\ \n(1) 求证: 数列$\\{b_n\\}$是等差数列;\\\\ \n(2) 若$a_1=1$, 求数列$\\{a_n\\}$的通项公式.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "解答题",
         "ans": "",
         "solution": "",
@@ -632522,9 +632559,11 @@
     },
     "023574": {
         "id": "023574",
-        "content": "某企业是用电大户, 去年的用电量达到$20$万度, 经预测, 在去年的基础上, 今年该企业若减少用电$x$万度, 今年的受损效益$S(x)$(万元)满足 $S(x)=\\begin{cases}  50x^2,  &1\\le x\\le 4, \\\\ 100x-\\dfrac{400}{x}+500, & 4<x\\le 20. \\end{cases}$ 为解决用电问题, 今年该企业决定进行技术升级, 实现效益增值, 今年的增效效益$Z(x)$(万元)满足$Z(x)=\\begin{cases}  \\dfrac{S(x)}{x},  &1\\le x\\le 4, \\\\ \\dfrac{S(x)-800}{x}+520, & 4<x\\le 20. \\end{cases}$  政府为鼓励企业节能, 补贴节能费用$n(x)=100x$万元. \\\\\n(1) 减少用电量多少万度时, 今年该企业增效效益恰好达到$544$万元; \\\\\n(2) 减少用电量多少万度时, 今年该企业总效益最大?（总效益$=$增效效益$+$补贴节能费用$-$受损效益）",
+        "content": "某企业是用电大户, 去年的用电量达到$20$万度, 经预测, 在去年的基础上, 今年该企业若减少用电$x$万度, 今年的受损效益$S(x)$(万元)满足 $S(x)=\\begin{cases}  50x^2,  &1\\le x\\le 4, \\\\ 100x-\\dfrac{400}{x}+500, & 4<x\\le 20. \\end{cases}$ 为解决用电问题, 今年该企业决定进行技术升级, 实现效益增值, 今年的增效效益$Z(x)$(万元)满足$Z(x)=\\begin{cases}  \\dfrac{S(x)}{x},  &1\\le x\\le 4, \\\\ \\dfrac{S(x)-800}{x}+520, & 4<x\\le 20. \\end{cases}$  政府为鼓励企业节能, 补贴节能费用$n(x)=100x$万元. \\\\\n(1) 减少用电量多少万度时, 今年该企业增效效益恰好达到$544$万元; \\\\\n(2) 减少用电量多少万度时, 今年该企业总效益最大? (总效益$=$增效效益$+$补贴节能费用$-$受损效益)",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第二单元"
+        ],
         "genre": "解答题",
         "ans": "",
         "solution": "",
@@ -632544,7 +632583,9 @@
         "id": "023575",
         "content": "已知数列$\\{a_n\\}$是等比数列,  $a_1=-\\dfrac{9}{4}$, 公比$q=\\dfrac{3}{4}$,  数列$\\{b_n\\}$满足$3b_n+(n-4)a_n=0$. 记$\\{b_n\\}$的前$n$项和为$T_n$. \\\\\n(1) 求数列$\\{a_n\\}$的通项公式及数列$\\{a_n\\}$前$n$项和$S_n$; \\\\\n(2) 用数学归纳法证明$T_n=-4n(\\dfrac{3}{4})^{n+1}$; \\\\\n(3) 若对任意的$n\\geq 1$,$n\\in \\mathbf{N}$, 均有$T_n\\leq m b_n$恒成立, 求实数$m$的取值范围.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "解答题",
         "ans": "",
         "solution": "",
@@ -632564,7 +632605,9 @@
         "id": "023576",
         "content": "设函数$f(x)$的定义域为$D$, 对于区间$I=[a,b]$($a<b$, $I\\subseteq D$), 若满足以下两条性质之一, 则称$I$为$f(x)$的一个“$\\Omega$区间”.\\\\\n性质1: 对任意$x\\in I$, 有$f(x)\\in I$;\\\\\n性质2: 对任意$x\\in I$, 有$f(x)\\notin I$.\\\\\n(1) 分别判断区间$[1,2]$是否是 $f(x)=3-x$和$g(x)=\\dfrac{3}{x}$的``$\\Omega$区间''(直接写出结论);\\\\\n(2) 若$[0,m]$($m>0$)是函数$f(x)=-x^2+2x$的``$\\Omega$区间'', 求$m$的取值范围;\\\\\n(3) 已知定义在$\\mathbf{R}$上且图像是一段连续曲线的函数$f(x)$满足: 对任意$x_1,x_2\\in \\mathbf{R}$, 且$x_1\\neq x_2$, 有$\\dfrac{f(x_2)-f(x_1)}{x_2-x_1}<-1$. 求证：$f(x)$存在``$\\Omega$区间'', 且存在$x_0\\in \\mathbf{R}$, 使得$x_0$不属于$f(x)$的所有``$\\Omega$区间''.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第二单元"
+        ],
         "genre": "解答题",
         "ans": "",
         "solution": "",
@@ -632585,7 +632628,9 @@
         "id": "023577",
         "content": "在等差数列 $\\{a_n\\}$ 中, $a_1=1$, 公差 $d=2$, 则 $a_3=$\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632618,7 +632663,9 @@
         "id": "023578",
         "content": "若 $\\mathrm{P}_n^2=n \\mathrm{P}_3^3$, 则 $n=$\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第八单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632651,7 +632698,9 @@
         "id": "023579",
         "content": "某医疗机构有 $4$ 名新冠疫情防控志愿者, 现要从这 $4$ 人中选 $3$ 个人去 $3$ 个不同的社区进行志愿服务. 则不同的选择办法共有\\blank{50}种.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第八单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632684,7 +632733,9 @@
         "id": "023580",
         "content": "已知圆锥的底面半径为 $1$ ,母线长为 $2$, 则该圆锥的侧面积为\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第六单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632717,7 +632768,9 @@
         "id": "023581",
         "content": "已知球的表面积为 $16 \\pi$, 则该球的体积为\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第六单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632750,7 +632803,9 @@
         "id": "023582",
         "content": "设 $(3 x-2)^4=a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4$, 则 $a_0+a_1+a_2+a_3+a_4=$\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第八单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632783,7 +632838,9 @@
         "id": "023583",
         "content": "在 $1,2,3,4,5,6$ 这 $6$ 个数字中任取 $2$ 个相加, 和是 $2$ 的倍数的概率是\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第八单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632816,7 +632873,9 @@
         "id": "023584",
         "content": "空间内 $7$ 个点, 若其中有且只有 $4$ 点共面, 但无 $3$ 点共线, 可组成\\blank{50}个四面体.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第八单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632849,7 +632908,9 @@
         "id": "023585",
         "content": "小明为了解自己每天花在体育锻炼上的时间 (单位: $\\min$), 连续记录了 $7$ 天的数据并绘制成如图所示的茎叶图, 则这组数据的第 $60$ 百分位数是\\blank{50}.\n\\begin{center}\n\\begin{tabular}{l|lll}4 & 2 & 7 & \\\\\n5 & 4 & 5 & 8 \\\\\n7 & 0 & & \\\\\n9 & 6 & &\n\\end{tabular}\n\\end{center}",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第九单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632882,7 +632943,9 @@
         "id": "023586",
         "content": "某学校为了获得该校全体高中学生的体育锻炼情况, 按照男、女生的比例分别抽样调查了 $55$ 名男生和 $45$ 名女生的每周锻炼时间. 通过计算得到男生每周锻炼时间的平均数为 $8$ 小时, 方差为 $6$; 女生每周锻炼时间的平均数为 $6$ 小时, 方差为 $8$. 根据所有样本的方差来估计该校学生每周锻炼时间的方差为\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第九单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632915,7 +632978,9 @@
         "id": "023587",
         "content": "对于任意正整数 $n$, 定义``$n$ 的双阶乘 $n !!$''如下:\n对于 $n$ 是偶数时, $n ! !=n \\times(n-2) \\times(n-4) \\times \\cdots \\times 6 \\times 4 \\times 2$;\n对于 $n$ 是奇数时, $n ! !=n \\times(n-2) \\times(n-4) \\times \\cdots \\times 5 \\times 3 \\times 1$.\n现有如下四个命题:\\\\\n\\textcircled{1} $(2021 ! !) \\cdot(2022 ! !)=2022 ! $;\\\\\n\\textcircled{2} $2022 ! !=2^{1011}\\cdot 1011 ! $;\\\\\n\\textcircled{3} $2022 ! !$ 的个位数是 $0$;\\\\\n\\textcircled{4} $2023 ! ! $ 的个位数是 $5$.\\\\\n正确的命题序号为\\blank{50}.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第八单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632948,7 +633013,9 @@
         "id": "023588",
         "content": "在《九章算术》中, 将底面为直角三角形, 侧棱垂直于底面的三棱柱称之为堑堵, 如图, 在堑堵 $ABC-A_1B_1C_1$ 中, $AB=BC$, $A_1A>AB$, 堑堵的顶点 $C_1$到直线 $A_1C$ 的距离为 $m, C_1$ 到平面 $A_1BC$ 的距离为 $n$, 则 $\\dfrac{n}{m}$ 的取值范围是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2-0.2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2+0.2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\draw [dashed] (A_1)--(C);\n\\draw (A_1)--(B);\n\\end{tikzpicture}\n\\end{center}",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第六单元"
+        ],
         "genre": "填空题",
         "ans": "",
         "solution": "",
@@ -632981,7 +633048,9 @@
         "id": "023589",
         "content": "若 $P(A \\cap B)=\\dfrac{1}{9}$, $P(\\overline{A})=\\dfrac{2}{3}$, $P(B)=\\dfrac{1}{3}$, 则事件 $A$ 与 $B$ 的关系是 \\bracket{20}.\n\\twoch{事件 $A$ 与 $B$ 互斥}{事件 $A$ 与 $B$ 对立}{事件 $A$ 与 $B$ 相互独立}{事件 $A$ 与 $B$ 既互斥又相互独立}",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第八单元"
+        ],
         "genre": "选择题",
         "ans": "",
         "solution": "",
@@ -633014,7 +633083,9 @@
         "id": "023590",
         "content": "如图, 在棱长为 $2$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, 点 $P$ 在截面 $A_1DB$上(含边界), 则线段 $AP$ 的最小值等于\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (A_1)--(B);\n\\draw [dashed] (A_1)--(D)--(B);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{2}{3}$}{$\\dfrac{2 \\sqrt{3}}{3}$}{$\\sqrt{2}$}{$\\dfrac{\\sqrt{3}}{3}$}",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第六单元"
+        ],
         "genre": "选择题",
         "ans": "",
         "solution": "",
@@ -633047,7 +633118,9 @@
         "id": "023591",
         "content": "在某区高三年级举行的一次质量检测中, 某学科共有 $3000$ 人参加考试. 为了解本次考试学生的成绩情况, 从中抽取了部分学生的成绩(成绩均为正整数, 满分为 $100$ 分)作为样本进行统计, 样本容量为 $n$. 按照 $[50,60)$、$[60,70)$、$[70,80)$、$[80,90)$、$[90,100]$ 的分组作出频率分布直方图(如图所示), 已知成绩落在 $[50,60)$ 内的人数为 $16$, 则下列结论正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (30,0) -- (36,0) -- (38,-0.002) -- (42,0.002) -- (44,0)--  (120,0) node [below] {成绩(分)};\n\\draw [->] (30,0) -- (30,0.05) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (30,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.016,60/0.03,70/0.04,80/0.01,90/0.004}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {50/0.016,60/0.03/x,70/0.04,80/0.01,90/0.004}\n{\\draw [dashed] (\\i,\\j) -- (30,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}\n\\onech{样本容量 $n=1000$}{图中 $x=0.025$}{若将该学科成绩由高到低排序, 前 $15 \\%$ 的学生该学科成绩为 A 等,则成绩为 78 分的学生该学科成绩肯定不是 A 等}{估计全体学生该学科成绩的平均分为 $70.6$ 分}",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第九单元"
+        ],
         "genre": "选择题",
         "ans": "",
         "solution": "",
@@ -633080,7 +633153,9 @@
         "id": "023592",
         "content": "已知等差数列 $\\{a_n\\}$ (公差不为 $0$) 和等差数列 $\\{b_n\\}$ 的前 $n$ 项和分别为 $S_n$、$T_n$, 如果关于 $x$ 的实系数方程 $1003 x^2-S_{1003}x+T_{1003}=0$ 有实数解, 那么以下 $1003$ 个方程 $x^2-a_i x+b_i=0 $($i=1,2, \\cdots 1003$) 中, 有实数解的方程至少有\\bracket{20}个.\n\\fourch{$499$}{$500$}{$501$}{$502$}",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "选择题",
         "ans": "",
         "solution": "",
@@ -633113,7 +633188,9 @@
         "id": "023593",
         "content": "如图, 已知点 $P$ 在圆柱 $OO_1$ 的底面圆 $O$ 上, $\\angle AOP=120^{\\circ}$, 圆 $O$ 的直径 $AB=4$, 圆柱的高 $OO_1=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw (-2,0) node [left] {$A$} coordinate (A) (2,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,3) node [left] {$A_1$} coordinate (A_1) (B) ++ (0,3) node [right] {$B_1$} coordinate (B_1);\n\\draw (0,0) node [above] {$O$} coordinate (O) (0,3) node [above] {$O_1$} coordinate (O_1);\n\\draw (A) arc (180:360:2 and 0.5) (A_1) arc (180:-180:2 and 0.5);\n\\draw (A)--(A_1)(B)--(B_1)(A_1)--(B_1);\n\\draw [dashed] (A)--(B)(A) arc (180:0:2 and 0.5);\n\\foreach \\i in {O,O_1}\n{\\filldraw (\\i) circle (0.05);};\n\\draw (-60:2 and 0.5) node [below] {$P$} coordinate (P);\n\\draw [dashed] (O)--(P)(A)--(P)--(B)(A_1)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆柱的表面积与体积;\\\\\n(2) 求直线 $A_1P$ 与 $AB$ 所成的角.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第六单元"
+        ],
         "genre": "解答题",
         "ans": "",
         "solution": "",
@@ -633146,7 +633223,9 @@
         "id": "023594",
         "content": "已知数列 $\\{a_n\\}$ 满足 $a_1=1$, $a_{n+1}=3 a_n+1 $($n \\geq 1$, $n \\in \\mathbf{N}$).\\\\\n(1) 求其通项公式 $a_n$;\\\\\n(2) 求数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "解答题",
         "ans": "",
         "solution": "",
@@ -633179,7 +633258,9 @@
         "id": "023595",
         "content": "(1) 求 $(1-\\dfrac{y}{x})^{10}$ 的二项展开式的中间项;\\\\\n(2) 若 $(1+\\dfrac{3}{x})^n=a_0+\\dfrac{a_1}{x}+\\dfrac{a_2}{x^2}+\\cdots+\\dfrac{a_n}{x^n}$, 且 $a_2=945$, 求 $a_i$($0 \\leq i \\leq n$, $i \\in \\mathbf{N}$) 中的最大值.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第八单元"
+        ],
         "genre": "解答题",
         "ans": "",
         "solution": "",
@@ -633212,7 +633293,10 @@
         "id": "023596",
         "content": "在 2019 中国北京世界园艺博览会期间, 某工厂生产 $A$、$B$、$C$ 三种纪念品, 每一种纪念品均有精品型和普通型两种, 某一天产量如下表: (单位: 个)\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 纪念品 $A$ & 纪念品 $B$ & 纪念品 $C$ \\\\\n\\hline 精品型 & $100$ & $150$ & $n$ \\\\\n\\hline 普通型 & $300$ & $450$ & $600$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n现采用分层抽样的方法在这一天生产的纪念品中抽取 $200$ 个, 其中 $A$ 种纪念品有 $40$ 个.\\\\\n(1) 求 $n$ 的值;\\\\\n(2) 用分层抽样的方法在 $C$ 种纪念品中抽取一个容量为 $5$ 的样木, 从样本中任取 $2$ 个纪念品, 求至少有 $1$ 个精品型纪念品的概率;\\\\\n(3) 从 $B$ 种精品型纪念品中抽取 $5$ 个, 其某种指标的数据分别如下: $x$、$y$、$10$、$11$、$9$,把这 $5$ 个数据看作一个总体, 其均值为 $10$, 方差为 $2$, 求 $|x-y|$ 的值.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第九单元",
+            "第八单元"
+        ],
         "genre": "解答题",
         "ans": "",
         "solution": "",
@@ -633245,7 +633329,9 @@
         "id": "023597",
         "content": "按照如下规则构造数表: 第一行是:  $2$; 第二行是: $2+1,2+3$; 即 $3,5$, 第三行是: $3+1,3+3,5+1,5+3$ 即 $4,6,6,8$; $\\cdots$ (即从第二行起将上一行的数的每一项各项加 $1$ 写出, 再各项加 $3$ 写出). 记第 $n$ 行所有的项的和为 $a_n$.\\\\\n\\begin{center}\n\\fbox{\\begin{tabular}{cccccccc}\n2 \\\\\n3& 5 \\\\\n4&6&6&8\\\\\n5&7&7&9&7&9&9&11\\\\\n\\multicolumn{8}{c}{$\\cdots\\cdots$}\n\\end{tabular}}\n\\end{center}\n(1) 求 $a_3, a_4, a_5, a_6$;\\\\\n(2) 试求 $a_{n+1}$ 与 $a_n$ 的递推关系, 并据此求出数列 $\\{a_n\\}$ 的通项公式;\\\\\n(3) 设 $S_n=\\dfrac{a_3}{a_1 a_2}+\\dfrac{a_4}{a_2 a_3}+\\cdots \\dfrac{a_{n+2}}{a_n a_{n+1}}$($n \\geq 1$, $n \\in \\mathbf{N}$), 求 $S_n$.",
         "objs": [],
-        "tags": [],
+        "tags": [
+            "第四单元"
+        ],
         "genre": "解答题",
         "ans": "",
         "solution": "",