Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $n$ 为不超过 2015 的正整数，且 $1^{n}+2^{n}+3^{n}+4^{n}$ 的个位数为 0 ，求满足条件的正整数 $n$ 的个数。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

空间中点集定义如下: $A_{n}=\left\{\left.(x, y, z) \in \mathbf{R}^{3}|3| x\right|^{n}+|8 y|^{n}+|z|^{n} \leq 1\right\}, A=\bigcup_{n=1}^{\infty} A_{n}$, 求由 $A$ 中的点组成的图形的体积。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $x+y+z=2016, \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2016}$, 求 $(x-2016)(y-2016)(z-2016)$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

正方体的 $8$ 个顶点中任取 $3$ 个构成三角形，求三角形是等腰三角形的概率。答案的形式为\frac{k}{m}，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

用 $a_{n}$ 表示距离 $\sqrt{n}$ 最近的正整数, 已知 $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=2016$, 求 $n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数, 其中两个根的和为 $2+i$, 另两根的积为 $5+6 i$,则 $b$ 等于多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在正方体的八个顶点中任取 3 个构成三角形，求所构成的三角形是等腰三角形的概率。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若方程 $x^{2}-3 x-1=0$ 的根也是方程 $x^{4}+a x^{2}+b x+c=0$ 的根, 则 $a+b-2 c$ 的值为多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 \(k \neq 1\)，则等比数列 \(a+\log _{2} k, a+\log _{4} k, a+\log _{8} k\) 的公比为多少？答案应为 \(\frac{m}{n}\) 的形式，请给出 \(m+n\) 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $A B, C D$ 是圆 $O$ 的两条垂直直径，弦 $D F$ 交 $A B$ 于点 $E, D E=24, E F=18$，求 $O E$ 的值。原始答案的形式为 $k\sqrt{m}$，请给出 $k + m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

函数 $f(x)=x(x+1)(x+2)(x+3)$ 的最小值为多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

整数 $p, q$ 满足 $p+q=218, x^{2}+p x+q=0$ 有整数根，求满足这样条件的整数对 $(p, q)$ 的个数。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知正整数 $n$ 满足 $n \neq 2017$ ，且 $n^{n}$ 与 $2017^{2017}$ 有相同的个位数字，则 $|2017-n|$ 的最小值为多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知存在正整数 $a, b, c$ 满足 $a+b+c=407,10^{n} \mid a b c$ ，则 $n$ 的最大值是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $a, b, c$ 和 $\left(a-\frac{1}{b}\right)\left(b-\frac{1}{c}\right)\left(c-\frac{1}{a}\right)$ 均为正整数，求 $2 a+3 b+5 c$ 的最大值和最小值之差。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一群学生参加学科夏令营，每名同学至少参加数学、物理、化学中的一门学科考试. 已知有 $100$ 名学生参加了数学考试，$50$名学生参加了物理考试， $48$ 名学生参加了化学考试.若学生总数是参加至少两门考试的学生的 $2$ 倍，也是参加三门考试学生的 $3$ 倍，则学生总数为多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若实数 $x, y$ 满足 $\frac{x^{2}}{4}+y^{2}=1$ ，则 $|3 x+4 y-12|$ 的取值范围是 $[c, d]$。请分别求出 $c$ 和 $d$ 的值，并给出 $c + d$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知三个复数 $z_{1}=(5-a)+(6-4 b) \mathrm{i}, z_{2}=(2+2 a)+(3+b) \mathrm{i}, z_{3}=(3-a)+(1+3 b) \mathrm{i}(a, b \in$ $\mathbf{R})$. 若 $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|$ 取最小值, 则 $3 a+6 b$ 的结果为 \frac{k}{m} 的形式，请给出 $k+m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=12, a_{n+1}=\frac{1}{4}\left(3+a_{n}+\right.$ $\left.3 \sqrt{1+2 a_{n}}\right)\left(n \in \mathbf{N}^{*}\right)$ ，则与 $a_{10}$ 最接近的整数是 ______。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若复数 \(z\) 满足 \(\frac{z}{2}\) 与 \(\frac{2}{z}\) 的实部、虚部均在区间 \([-1,1]\) 上取值，则 \(z\) 在复平面上形成轨迹的面积是多少？答案的形式为 \(a-b\pi\)，请给出\(a+b\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

用 $[x]$ 表示不超过 $x$ 的最大整数，比如 [1.2] $=1$, $[-1.2]=-2$. 若 $\alpha=\frac{1+\sqrt{5}}{2}$, 求 $\left[\alpha^{12}\right]$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若二次函数 $f(x)$ 满足 $\forall x \in \mathbf{R}, 2 x \leqslant f(x) \leqslant \frac{x^{2}+4}{2}$ 及 $f(-2)=0$ ，求 $f(10)$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

满足正整数 $n \leqslant 2022$ 且 $2n+1$ 与 $3n+1$ 均为完全平方数的 $n$ 的个数是 _______.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $\triangle A B C$ 的三边长分别是 $2,3,4$, 其外心为 $O$, 求 $\overrightarrow{O A} \cdot \overrightarrow{A B}+\overrightarrow{O B} \cdot \overrightarrow{B C}+$ $\overrightarrow{O C} \cdot \overrightarrow{C A}$ 的值。答案的形式为\frac{k}{m}，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在长方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 中，$A B=2, A D=A A_{1}=1$，求点 $A$ 到平面 $A_{1} B D$ 的距离。答案的形式为\frac{k}{m}，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若从 $0,1,2, \cdots, 9$ 中选取 $5$ 个两两互异的数字依次排成一个五位数（包括 $0$ 在首位的五位数，其大小就是把 $0$ 去掉后的四位数），求它能被 $396$ 整除的概率。答案的形式为\frac{a}{b}，请给出a+b的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若空间向量 $\boldsymbol{a} ,\boldsymbol{b} ,\boldsymbol{c}$ 满足 $|\boldsymbol{a}| \leqslant 1,|\boldsymbol{b}| \leqslant 1, \mid \boldsymbol{a}+2 \boldsymbol{b}+$ $\boldsymbol{c}|=|\boldsymbol{a}-2 \boldsymbol{b}|$ ，求 $| \boldsymbol{c} \mid$ 的最大值。答案的形式为 $k\sqrt{m}$，请给出 $k+m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $x^{2}+y^{2} \leqslant 1(x y \in \mathbf{R})$, 求 $x^{2}+x y-y^{2}$ 的最大值与最小值所组成的区间。答案形如 \left[-\frac{\sqrt{k}}{m}, \frac{\sqrt{k}}{m}\right]，请给出k + m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $A, B, C \subseteq\{1,2,3, \cdots, 2020\}, A \subseteq C, B \subseteq C$ ，求有序集合组 $(A, B, C)$ 的组数。原始答案的形式为 $k^m$，请给出 $k + m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Evaluate \(\lim \limits_{n \rightarrow \infty} \sum\limits_{k=1}^{n} \arctan \frac{2}{k^{2}}\). The answer is in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

函数 $f(x)=\frac{2 e^{x}}{e^{x}+e^{-x}}+\sin x(-2 \leqslant x \leqslant 2)$ 的最大值与最小值之和是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $P(X=k)=\frac{1}{2^{k}}(k=1,2,3, \cdots)$，若 $Y$ 为 $X$ 除以 $3$ 所得的余数，则求随机变量 $Y$ 的期望。原始答案的形式为 \frac{k}{m}，请给出 $k + m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若某个正四棱锥的相邻两个侧面所成二面角的大小为 \(\alpha\)，侧棱与底面所成线面角的大小为 \(\beta\)，求满足条件的表达式的值。答案应为 \(k \sec \alpha + m \tan^2 \beta = n\) 的形式，请给出 \(k + m + n\) 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若函数 $f(t)=t^{2}+2 t$ ，则点集 $\left\{(x, y) \left\lvert\,\left\{\begin{array}{l}f(x)+f(y) \leqslant 2 \\ f(x) \geqslant f(y)\end{array}\right\}\right.\right.$ 所构成图形的面积是 $k\pi$，求 $k$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $a^{2}(b+c)=b^{2}(a+c)=1(a \neq b)$ ，则 $(a+b) c^{2}-a b c$ 的值为多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

顶点为 $P$ 的圆锥的轴截面是等腰直角三角形， $A$ 是底面圆周上的点， $B$ 是底面圆内的点， $O$ 为底面圆圆心， $A B \perp O B$ ，垂足为 $B, O H \perp P B$ ，垂足为 $H$ ，且 $P A=4, C$ 为 $P A$ 的中点，求当三棱锥 $O-H P C$ 的体积最大时， $O B$ 的长。答案的形式为 \frac{k \sqrt{m}}{n}，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求不等式 $\sqrt{\log _{2} x-1}+\frac{1}{2} \log _{\frac{1}{2}} x^{3}+2>0$ 的解集。解集的形式为 [a, b)，请给出 a + b 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

记集合 $T=\{0,1,2,3,4,5,6\}, M=\left\{\left.\frac{a_{1}}{7}+\frac{a_{2}}{7^{2}}+\frac{a_{3}}{7^{3}}+\frac{a_{4}}{7^{4}} \right\rvert\, a_{i} \in T, i=1,2,3,4\right\}$ ，将 $M$ 中的元素按从大到小排列，求第 2005 个数。原始答案的形式为 \frac{b_1}{7}+\frac{b_2}{7^{2}}+\frac{b_3}{7^{3}}+\frac{b_4}{7^{4}}，请给出b_1 + b_2 + b_3 + b_4的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

使关于 $x$ 的不等式 $\sqrt{x-3}+\sqrt{6-x} \geqslant k$ 有解的实数 $k$ 的最大值是多少？请以$\sqrt{m}$的形式给出答案，并求出m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 $a$ 使得不等式 $|2 x-a|+|3 x-2 a| \geqslant a^{2}$ 对任意实数 $x$ 恒成立，求满足条件的 $a$ 所组成的集合的区间端点。如果区间为 $[m, n] = \left[\frac{k}{l}, \frac{p}{q}\right]$，请给出k+l+p+q的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

以 $\sqrt{2}$ 和 $1-\sqrt[3]{2}$ 为两个根的有理系数多项式的次数最低是____。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $x, y \in \mathbf{Z}$ ，若 $\left(x^{2}+x+1\right)^{2}+\left(y^{2}+y+1\right)^{2}$ 为完全平方数，则数对 $(x, y)$ 的对数是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$1 \times 1!+2 \times 2!+\cdots+672 \times 672!$ 被 $2019$ 除的余数是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$n$ 是任意自然数，$1^{3}+2^{3}+3^{3}+\cdots+n^{3}$ 的个位数不可能是______。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 \(x, y, z>0\)，则 \(f(x, y, z)=\frac{x y z}{(4 x+1)(9 x+y)(4 y+z)(z+9)}\) 的最大值为多少？答案的形式为 \(\frac{k}{m^2}\)，请给出 \(k + m\) 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$f(x)=\frac{x}{\sqrt{1+x^{2}}}+\frac{1-x^{2}}{1+x^{2}}$ 的取值范围是 $(-2, y]$，其中$y$的原始答案是以\frac{k}{m}的形式给出，请计算k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设非负实数 $x, y$ 满足 $2 x+y=1$, 求 $x+\sqrt{x^{2}+y^{2}}$ 的最小值。答案的形式为\frac{k}{m}，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $a_{1}, a_{2}, a_{3}, a_{4} \in\{1,2,3,4\}, N\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ 为 $a_{1}, a_{2}, a_{3}, a_{4}$ 中不同数字的种类，如 $N(1,1,2,3)=3, N(1,2,2,1)=2$ ，则所有的 $a_{1}, a_{2}, a_{3}, a_{4}$ 的排列 （共 $4^{4}=256$ 个）所得的 $N\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ 的平均值的答案是一个分数。请将其表示为\frac{k}{m}的形式，并给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在四面体 $P A B C$ 中, 若 $\triangle A B C$ 是边长为 3 的正三角形, 且 $P A=3, P B=4$, $P C=5$ ，求该四面体的体积。答案的形式为\(\sqrt{k}\)，请给出k的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 \(\triangle ABC\) 中，求 \(\sin A + \sin B \sin C\) 的最大值。答案的形式为 \(\frac{k + \sqrt{m}}{n}\)，请计算k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

不定方程 $x+2 y+3 z=100$ 的非负整数解的组数是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $\omega=\cos \frac{2 \pi}{5}+\operatorname{isin} \frac{2 \pi}{5}, f(x)=x^{2}+x+2$, 求 $f(\omega) f\left(\omega^{2}\right) f\left(\omega^{3}\right) f\left(\omega^{4}\right)$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 \(z, w \in \mathbf{C},|z+w|=1,\left|z^{2}+w^{2}\right|=4\)，求 \(|z w|\) 的最大值。原始答案的形式为 \(\frac{k}{m}\)，请给出\(k+m\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设复数 \(z, w\) 满足: \(|w+z|=1,\left|w^{2}+z^{2}\right|=4\)，求 \(|w z|\) 的最小值。答案应以 \(\frac{k}{m}\) 的形式给出，其中 \(\frac{k}{m}\) 为最简分数。请计算 \(k+m\) 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $A, B, C$ 是三角形的三个内角，求 $\sin A+\sin B \sin C$ 的最大值。答案的形式为 \(\frac{k+\sqrt{n}}{m}\)，请给出k + n + m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $\vec{e}_{1}, \vec{e}_{2}$ 是两个单位向量, $x, y$ 是实数. 若 $\left\langle\vec{e}_{1}, \vec{e}_{2}\right\rangle=\frac{\pi}{3},\left|x \vec{e}_{1}+y \vec{e}_{2}\right|=1$, 求 $x$ 的最大值。答案的形式为 \frac{k \sqrt{n}}{m}，请给出k+m+n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知集合 $M=\{-1,0,1\}, N=\{2,3,4,5,6\}$ 。设映射 $f: M \rightarrow N$ 满足：对任意的 $x \in M, x+f(x)+x f(x)$ 是奇数。求这样的映射 $f$ 的个数。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 Rt \(\triangle A B C\) 中, \(\angle A B C=\frac{\pi}{2}, A B=\sqrt{3}, B C=1, \frac{\overrightarrow{P A}}{|\overrightarrow{P A}|}+\frac{\overrightarrow{P B}}{|\overrightarrow{P B}|}+\frac{\overrightarrow{P C}}{|\overrightarrow{P C}|}=\mathbf{0}\)。若\(\angle A P B = x \pi\)，原始答案的形式为\(\frac{k}{m}\)，请给出\(k + m\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设复数 \(z\) 满足 \(|3z-7i|=3\)，求 \(\left|\frac{z^{2}-2z+2}{z-1+i}\right|\) 的最大值。答案的形式为 \(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将号码分别为 \(1, 2, \cdots, 9\) 的九个小球放入一个袋中, 这些小球仅号码不同, 其余完全相同。甲从袋中摸出一个小球，其号码为 \(a\)，放回后，乙从此袋中再摸出一个球，其号码为 \(b\)。求使得不等式 \(a-2b+10>0\) 成立的事件发生的概率。答案的形式为 \(\frac{k}{m}\)，请给出 \(k+m\) 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

甲、乙两人进行乒乓球比赛，约定每局胜者得 1 分，负者得 0 分，比赛进行到有一人比对方多 2 分或打满 6 局时停止。设甲在每局中获胜的概率为 \(\frac{2}{3}\)，乙在每局中获胜的概率为 \(\frac{1}{3}\)，且各局胜负相互独立。求比赛停止时已打局数 \(\xi\) 的期望 \(E \xi\)。答案形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

函数 $f(x)=\frac{5-4 x+x^{2}}{2-x}$ 在 $(-\infty, 2)$ 上的最小值是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知函数 \(f(x)=x^{2}-2 t x+t\)。当 \(x \in [-1,1]\) 时，记 \(f(x)\) 的最小值为 \(m\)。求 \(m\) 的最大值。原始答案的形式为\(\frac{k}{m}\)，请给出\(k+m\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

对任意正整数 $n$ 与 $k(k \leqslant n), f(n, k)$ 表示不超过 $\left[\frac{n}{k}\right]$ （ $[x]$ 表示不超过实数 $x$ 的最大整数）且与 $n$ 互素的正整数的个数。求 $f(100,3)$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $(x+y+z)^{8}$ 的展开式中，求所有形如 $x^{2} y^{a} z^{b}(a, b \in \mathbf{N})$ 的项的系数之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 \(\triangle A B C\) 中, \(\overrightarrow{A B} \cdot \overrightarrow{B C}=3 \overrightarrow{C A} \cdot \overrightarrow{A B}\)。求 \(\frac{|\overrightarrow{A C}|+|\overrightarrow{A B}|}{|\overrightarrow{B C}|}\) 的最大值。答案的形式为\(\sqrt{m}\)，请给出m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $(x+y+z)^{8}$ 的展开式中，所有形如 $x^{2} y^{a} z^{b}(a 、 b \in \mathbf{N})$ 的项的系数之和为多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $F_{1} 、 F_{2}$ 为椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的左、右焦点，该椭圆上存在两点 $A 、 B$ ，使得 $\overrightarrow{F_{1} A}=3 \overrightarrow{F_{2} B}$. 求该椭圆的离心率的取值范围 $(m, n)$，其中 $0 < m < n < 1$。答案形式为 $(\frac{k}{l}, p)$，其中 $k$ 和 $l$ 互质且 $l$ 不能为零。请给出 $k + l + p$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知数列 $\left\{a_{n}\right\}$ 满足
$a_{n}=(\sqrt{2}+1)^{n}-(\sqrt{2}-1)^{n}(n \in \mathbf{N})$.
记 $[x]$ 表示不超过实数 $x$ 的最大整数.求 $\left[a_{2017}\right]$ 的个位数字。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知方程 $1+x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots-\frac{x^{2018}}{2018}=0$ 的所有实数根都在区间 $[a, b]$ 内（其中 $a, b \in \mathbf{Z}$,且 $a<b$ ), 求 $b-a$ 的最小值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$A B$ 为过抛物线 $y^{2}=4 x$ 焦点 $F$ 的弦, $O$ 为坐标原点, 且 $\angle O F A=135^{\circ}, C$ 为抛物线准线与 $x$ 轴的交点，求 $\angle A C B$ 的正切值。答案的形式为 $k \sqrt{m}$，请给出 k + m 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设复数 $z_{1}, z_{2}$ 在复平面内对应的点分别为 $Z_{1}, Z_{2}, O$ 为坐标原点, 若 $\left|z_{1}\right|=1$, $5 z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}=0$, 求 $\triangle O Z_{1} Z_{2}$ 的面积。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求值: \(\lim _{n \rightarrow+\infty}\left(\sum_{k=1}^{n} \arctan \frac{2}{k^{2}}\right)=(\quad)\pi\)。如果答案是\(\frac{k}{m}\pi\)，请填写\(k+m\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 $x_{1}, x_{2}, \cdots, x_{21}$ 满足 $0 \leqslant x_{i} \leqslant 1(1,2, \cdots, 21)$, 求 $\sum_{i=1}^{21} \sum_{k=1}^{21}\left|x_{i}-x_{k}\right|$ 的最大值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 $x_{1}, x_{2}, \cdots, x_{21}$ 满足 $0 \leq x_{i} \leq 1(1,2, \cdots, 21)$ ，求 $\sum_{i=1}^{21} \sum_{k=1}^{21}\left|x_{i}-x_{k}\right|$ 的最大值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设复数 $z$ 满足 $|3 z-7 \mathrm{i}|=3$, 求 $\left|\frac{z^{2}-2 z+2}{z-1+\mathrm{i}}\right|$ 的最大值。答案的形式为 $\frac{k}{m}$，请给出 $k + m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $A, B, C$ 是集合 $\{1,2, \cdots, 2020\}$ 的子集，且满足 $A \subseteq B \subseteq C$ ，求这样的有序组 $(A, B, C)$ 的总数。答案的形式为 $x^y$，请给出x和y的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There exist positive integers $A$, $B$, and $C$, with no common factor greater than 1, such that:
\[A \log_{200} 5 + B \log_{200} 2 = C. \]
Find the value of $A + B + C$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie, Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Six distinct integers are picked at random from \(\{1,2,3,\ldots,10\}\). What is the probability that, among those selected, the second smallest is \(3\)? Express your answer in the form of \(\frac{k}{m}\), where the fraction is in simplest form, and find the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Triangle \( ABC \) has side lengths in arithmetic progression, and the smallest side has length \( 6 \). If the triangle has an angle of \( 120^\circ \), what is the area of \( ABC \)? The original answer is in the form of \( k\sqrt{m} \). Please provide the value of \( k + m \).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let the sequence \((a_n)\) be defined as \(a_1=\frac{1}{3}\) and \(a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}\) for every \(n\geq 1\). If \(a_k\) is the largest term of the sequence satisfying \(a_k < \frac{1}{50}\), find the value of \(k\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the figure, $ABCD$ is a $2 \times 2$ square, $E$ is the midpoint of $\overline{AD}$, and $F$ is on $\overline{BE}$. If $\overline{CF}$ is perpendicular to $\overline{BE}$, find the area of quadrilateral $CDEF$. The original answer is in \frac{k}{m} format, where k and m are coprime integers. Please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The cost $C$ of sending a parcel post package weighing $P$ pounds, where $P$ is an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. What is the formula for the cost $C$ in terms of $P$? The original answer is in the form $C = a + b(P - 1)$, please give the value of $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A circular piece of metal of maximum size is cut out of a square piece, and then a square piece of maximum size is cut out of the circular piece. Find the total amount of metal wasted as a fraction of the area of the original square. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let \((1+x+x^2)^n = a_0 + a_1x + a_2x^2 + \ldots + a_{2n}x^{2n}\) be an identity in \(x\). If we let \(s = a_0 + a_2 + a_4 + \ldots + a_{2n}\), the original answer is in the form \(\frac{k \cdot 3^n + m}{n}\), where n cannot be divisible by 2. Please find the value of \(k + m + n\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20, 20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

定义序列 $\{a_n\}_{n\geq 1}$ 如下：
- 当 $n \leq 2$ 时，$a_n = n - 1$
- 对于所有 $n \geq 2$，$a_n$ 是 $a_{n-1} + a_{n-2}$ 除以 $3$ 的余数

计算从 $a_{2018}$ 到 $a_{2025}$ 的项的和 $\sum_{i=2018}^{2025} a_i$。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $P(x) = 1 - x + x^2 - x^3 + \dots + x^{18} - x^{19}$ and $Q(x) = P(x - 1)$. Find the coefficient of $x^2$ in polynomial $Q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Square $BDEC$ with center $F$ is constructed on the outside of triangle $ABC$ such that $\angle A = 90^\circ$, $|AB| = \sqrt{12}$, $|AC| = 2$. If $[AF] \cap [BC] = \{G\}$, find the length of $|BG|$. The original answer is in the form $m - n\sqrt{p}$, please give the value of $m + n + p$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many ways are there to make change for $55$ cents using any number of pennies, nickels, dimes, and quarters?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A six-digit number (base 10) is considered "squarish" if it satisfies the following conditions:

1. None of its digits are zero.
2. It is a perfect square.
3. The first two digits, the middle two digits, and the last two digits of the number are all perfect squares when considered as two-digit numbers.

How many squarish numbers are there? Provide your answer as an integer.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. Find the area of the region in the plane covered by these squares.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A pyramid of non-negative integers is constructed as follows:

1. The first row consists of only $0$.
2. The second row consists of $1$ and $1$.
3. The $n^{th}$ row (for $n > 2$) is an array of $n$ integers, where the leftmost and rightmost elements are equal to $n - 1$, and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row.

Let $S_n$ be the sum of numbers in the $n^{th}$ row. Determine the remainder when dividing $S_{2018}$ by $2018$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two-hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchantman makes $15$. At what time will the privateer overtake the merchantman? Provide your answer in the format of H:MM PM, where H and MM are two-digit numbers (possibly with leading zeros). Please give the value of H + MM.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

An observer stands on the side of the front of a stationary train. When the train starts moving with constant acceleration, it takes $5$ seconds for the first car to pass the observer. If the time it takes for the $10^{\text{th}}$ car to pass is $x$ seconds, find the value of $100x$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A box contains 11 balls, numbered 1, 2, 3, ..., 11. If 6 balls are drawn simultaneously at random, find the probability that the sum of the numbers on the balls drawn is odd. Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and provide the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If two poles $20''$ and $80''$ high are $100''$ apart, find the height (in inches) of the intersection of the lines joining the top of each pole to the foot of the opposite pole.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f(x) = \sqrt{\sin^4 x + 4\cos^2 x} - \sqrt{\cos^4 x + 4\sin^2 x}$. Find an equivalent form of $f(x)$ in the format $k\cos mx$. Please provide the value of $k + m$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many integer pairs \((x, y)\) are there such that:
\[0 \leq x < 165, \quad 0 \leq y < 165 \text{ and } y^2 \equiv x^3 + x \pmod{165}?\]
Find the number of such integer pairs.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a class, some pupils learn German, while others learn French. The number of girls learning French and the number of boys learning German total to 16. There are 11 pupils learning French, and there are 10 girls in the class. Besides the girls learning French, there are 16 pupils. How many pupils are there in the class?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $A$, $B$ be the number of digits of $2^{1998}$ and $5^{1998}$ in the decimal system. Find the value of $A + B$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white, and $2$ blue balls? Provide your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Calculate the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a ten-mile race, First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third? Provide your answer in the form of a fraction \frac{k}{m}, and then calculate the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Vessel $A$ contains liquids $X$ and $Y$ in the ratio $X:Y = 8:7$. Vessel $B$ holds a mixture of $X$ and $Y$ in the ratio $X:Y = 5:9$. In what ratio should you mix the liquids from both vessels to achieve a mixture with the ratio $X:Y = 1:1$? Express your answer in the form of $a:b$, and provide the value of $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of $50$ miles per hour, and Joe can run at a speed of $10$ miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. If the number of seconds he has to spare when he gets to the tunnel entrance is $x$, find the value of $10x$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?

\[
\begin{array}{cccc}
& 6 & 4 & 1 \\
& 8 & 5 & 2 \\
+ & 9 & 7 & 3 \\
\hline
2 & 4 & 5 & 6
\end{array}
\]

Provide the largest digit that can be changed.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f(x)=x^3+7x^2+9x+10$. Find the value of $p$ such that for every integer $a,b$, the statement
\[
f(a) \equiv f(b) \ (\text{mod } p) \Rightarrow a \equiv b \ (\text{mod } p)
\]
is satisfied.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If $\frac{m}{n} = \frac{4}{3}$ and $\frac{r}{t} = \frac{9}{14}$, find the value of $\frac{3mr - nt}{4nt - 7mr}$. The original answer is in \frac{a}{b} format, please give the value of a + b.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A watch loses $2\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March 21, what is the value of $n$ in the form of an improper fraction \frac{k}{m}? Please provide the sum of k and m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6 + 8 = 14$. Find the value of $E(1) + E(2) + E(3) + \cdots + E(100)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知四面体 \(D-A B C\) 中, \(A C=B C=A D=B D=1\)，求 \(D-A B C\) 体积的最大值。答案的形式为 \(\frac{k \sqrt{n}}{m}\)，请给出k+m+n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若正整数 $(m, n)$ 的最大公约数、最小公倍数分别是 $10!, 50!$ ，则数对 $(m, n)$ 的对数是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系中，$O$ 是坐标原点，若两定点 $A, B$ 满足 $|\overrightarrow{O A}|=|\overrightarrow{O B}|=\overrightarrow{O A} \cdot \overrightarrow{O B}=2$, 则点集 $\{P|\overrightarrow{O P}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B},|\lambda|+|\mu| \leqslant 1 \mu \in \mathbf{R}\}$ 所表示的区域的面积是多少？答案的形式为$k \sqrt{m}$，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$a, b, c$ 是等差数列，点 $P$ 到直线 $l: a x+b y+c=0$ 的垂足为 $M, P(-1,0), N(2,1)$，求 $M N$ 的最大值。答案的形式为 $k \sqrt{2}$，请给出 $k$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 2023 可以拆分为几个正整数之和，所有整数中的最大值和最小值相差不超过 1 的情况有多少种可能？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在正四面体 $A B C D$ 中，棱长为 $2 \sqrt{2}$。点 $P$ 满足 $|\overrightarrow{P A}+\overrightarrow{P B}|=2$。求 $\overrightarrow{A P} \cdot \overrightarrow{A D}$ 的最大值。答案的形式为 $k + m\sqrt{n}$，请给出 $k + m + n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

从棱长为 1 个单位长度的正方体的底面一顶点 $A$ 出发，每次均随机沿一条棱行走一个单位长度。进行 2 次这样的操作回到 $A$ 的概率是多少？答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $F\left(\frac{1-x}{1+x}\right)=x$ ，则满足该条件的等式可以表示为 F(kx+m) = n-F(x)。请计算k+m+n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $F\left(\frac{1-x}{1+x}\right)=x$ ，求出满足该条件的$F(x)$的性质，并填入下列等式中正确的表达式。原始答案的形式为F(-a-x) = -b - F(x)，请给出a + b的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 $a 、 b 、 c$ 满足
$$
\left\{\begin{array}{l}
a^{2}-b c-8 a+7=0 \\
b^{2}+c^{2}+b c-6 a+6=0
\end{array}\right.
$$
那么，求 $a$ 的取值范围。答案的形式为[a, b]，请给出a+b的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设有三个函数，第一个是 $y=\varphi(x)$ ，它的反函数是第二个函数，而第三个函数的图象与第二个函数的图象关于 $x+y=0$ 对称。求第三个函数的表达式。答案的形式为 $y = a\varphi(bx)$，请给出 $a + b$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知原点在粗圆 $k^{2} x^{2}+y^{2}-4 k x+2 k y+k^{2}-1=0$ 的内部，那么参数 $k$ 的取值范围是$-a < k < 0$ 或 $0 < k < b$。请计算a和b的绝对值之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 n 为不超过 2015 的正整数且 $1^{n}+2^{n}+3^{n}+4^{n}$ 的个位数为 0，求满足条件的正整数 n 的个数。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

三角形 \(A B C\) 的三个顶点分别对应复数 \(z_{1}, z_{2}, z_{3}\)，已知 \(\frac{z_{2}-z_{1}}{z_{3}-z_{1}}=1+2 i\)，求三角形 \(A B C\) 的面积与其最长边长的平方的比。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数，其中两个根的和为 $2+i$，另两根的积为 $5+6i$，求 $b$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $A$ 是以 $B C$ 为直径的圆上的一点, $D, E$ 是线段 $B C$ 上的点, $F$ 是 $B C$ 延长线上的点, 已知 $B F=4, B D=2, B E=5, \angle B A D=\angle A C D, \angle B A F=\angle C A E$, 求 $B C$ 的长。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $A B, C D$ 是圆 $O$ 的两条垂直直径, 弦 $D F$ 交 $A B$ 于点 $E, D E=24, E F=18$, 求 $O E$的值。答案的形式是$k \sqrt{m}$，请给出k + m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知正方形 \(A B C D\) 的边长为 \(1\)，\(P_{1}, P_{2}, P_{3}, P_{4}\) 是正方形内部的 4 个点，使得 \(\triangle A B P_{1}, \triangle B C P_{2}, \triangle C D P_{3}\) 和 \(\triangle D A P_{4}\) 都是正三角形。求四边形 \(P_{1} P_{2} P_{3} P_{4}\) 的面积。答案的形式为\(a-b\sqrt{c}\)，请给出a+b+c的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

正整数 $9+95+995+\cdots+\underbrace{99 \cdots 9}_{2016} 5$ 的十进制表示中数值 1 的个数为多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将等差数列 $1,5,9,13, \ldots, 2017$ 排成一个大数 $15913 \ldots 2017$, 则该数被 9 除的余数为多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知存在正整数 a, b, c 满足 $a+b+c=407,10^{n} \mid a b c$ ，求 n 的最大值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, $A=90^{\circ}, A B=1, A C=\sqrt{3}$, 点 P 满足 $\frac{\overrightarrow{P A}}{|\overrightarrow{P A}|}+\frac{\overrightarrow{P B}}{|\overrightarrow{P B}|}+\frac{\overrightarrow{P C}}{|\overrightarrow{P C}|}=\mathbf{0}$。如果答案是 $x^\circ$, 求 $x$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 \(\alpha, \beta\) 为锐角, 且 \(\cos (\alpha+\beta)=\frac{\sin \alpha}{\sin \beta}\), 求 \(\tan \alpha\) 的最大值。答案的形式为\(\frac{\sqrt{k}}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $A, B$ 分别是 $x$ 轴, $y$ 轴上的动点, 若以 $A B$ 为直径的圆 $C$ 与直线 $2x+y-4=0$ 相切, 则圆 $C$ 面积的最小值为 $\frac{k\pi}{m\pi}$，请给出 $k + m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知数列 \(\left\{a_{n}\right\}\) 的前 \(n\) 项和 \(S_{n}=(-1)^{n} \cdot a_{n}+\frac{1}{2^{n}}+n-3\)，且实数 \(t\) 满足 \(\left(t-a_{n}\right)\left(t-a_{n+1}\right)<0\)，求 \(t\) 的取值范围 \((\frac{a}{b}, \frac{c}{d})\)，其中 \(a < b\) 且 \(\frac{a}{b} < \frac{c}{d}\)。请给出a, b, c, d的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

\(\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} \arctan \frac{2}{k^{2}}\) 等于 \(x \pi\)，其中答案以\(\frac{k}{m}\)的最简分数形式给出，请求出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将函数 \(y=\sqrt{4+6 x-x^{2}}-2(x \in[0,6])\) 的图象逆时针方向旋转 \(\theta(0 \leqslant \theta \leqslant \alpha)\), 得到曲线 \(C\). 若对于每一个旋转角 \(\theta\)，曲线 \(C\) 都是一个函数的图象，则求 \(\alpha\) 的最大值。原始答案的形式为 \(\arctan \frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将函数 $y=\sqrt{4+6 x-x^{2}}-2(x \in[0,6])$ 的图象逆时针方向旋转 $\theta(0 \leqslant \theta \leqslant \alpha)$, 得到曲线 C。若对于每一个旋转角 $\theta$，曲线 C 都是一个函数的图象，则 $\alpha$ 的最大值为多少？原始答案的形式为\arctan \frac{k}{m}，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 \(y^{2}=4x\)，过 \(A(-2,3)\) 做抛物线两条切线，交 y 轴于 \(B, C\) 两点，求 \(\triangle ABC\) 的外接圆方程。答案的形式为\((x + \frac{k}{2})^2 + (y - \frac{m}{2})^2 = \frac{n}{p}\)，请给出k + m + n + p的值，其中\(\frac{n}{p}\)为最简分数。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

恰有一个实数 $x$ 使得 $x^{3}-a x-1=0$ 成立，求实数 $a$ 的取值范围。答案的形式为 \((-∞, \frac{a \sqrt[3]{b}}{c})\)，请给出 \(a+b+c\) 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将函数 \(y=\sqrt{4+6 x-x^{2}}-2(x \in[0, 6])\) 的图象绕坐标原点逆时针方向旋转角 \(\theta(0 \leqslant \theta \leqslant \alpha)\)，得到曲线 \(C\)。若对于每一个旋转角 \(\theta\)，曲线 \(C\) 都是一个函数的图象，则 \(\alpha\) 的最大值为多少？答案的形式为\(\arctan \frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知函数 \(f(x)=\sin x \cos x+\sin x+\frac{2}{5} \cos x\left(0 \leqslant x \leqslant \frac{\pi}{2}\right)\) 的最大值与最小值分别是 \(M, m\)，求 \(m\) 的值。答案的形式为 \(\frac{k}{m}\)，请给出 \(k+m\) 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在四面体 \(A B C D\) 中，若 \(A C = B C = A D = B D = 1\)，则求四面体 \(A B C D\) 体积的最大值。原始答案为\(\frac{k \sqrt{m}}{n}\)的形式，请给出k+m+n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 2023 可以拆分为几个正整数之和，所有整数中的最大值和最小值相差不超过 1。有多少种可能？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

圆周上 \(A_{1}, A_{2}, \cdots, A_{7}\) 七个点两两相连。求任选两条线段，这两条线段无公共点的概率。答案的形式为\(\frac{k}{m}\)，请计算k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设复数 $\omega=\left(\frac{a+\mathrm{i}}{1+\mathrm{i}}\right)^{2}$ ，其中 $a$ 为实数，若 $\omega$ 的实部为 2 ，求 $\omega$ 的虚部。答案的形式为 $-\frac{k}{m}$，请给出 $k+m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在四棱锥 $P-A B C D$ 中, $B_{1}, D_{1}$ 分别为侧棱 $P B, P D$ 的中点。求四面体 $A B_{1} C D_{1}$ 的体积与四棱锥 $P-A B C D$ 的体积之比。假设比值的形式为k:m，请计算k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 \(\triangle ABC\) 中，三边长 \(a, b, c\) 满足 \(a+c=3b\)，求 \(\tan \frac{A}{2} \tan \frac{C}{2}\) 的值。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设双曲线 $C_{1}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{4}=k(a>2, k>0)$, 椭圆 $C_{2}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$, 若 $C_{2}$ 的短轴长与 $C_{1}$ 的实轴长的比值等于 $C_{2}$ 的离心率，则求 $C_{1}$ 在 $C_{2}$ 的一条准线上截得线段的长。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $A+B=\frac{2 \pi}{3}$，求 $\cos ^{2} A+\cos ^{2} B$ 的最小值和最大值。答案分别为\frac{k}{m}和\frac{n}{p}的形式，请计算k+m+n+p的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $0<x<\frac{\pi}{2}$, 且 $\frac{\sin ^{4} x}{9}+\frac{\cos ^{4} x}{4}=\frac{1}{13}$, 则 $\tan x$ 的值可以表示为\frac{k}{m}的形式，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $0<k<4$, 直线 $l_{1}: k x-2 y-2 k+$ $8=0$ 和直线 $l_{2}: 2 x+k^{2} y-4 k^{2}-4=0$ 与两坐标轴围成一个四边形。求使这个四边形面积最小的 $k$ 值。原始答案是以\(\frac{a}{b}\)的形式给出的，请给出a+b的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 5 件产品中有 4 件正品、1 件次品。从中任取两件, 记其中含正品的个数为随机变量 \(\xi\)。求 \(\xi\) 的数学期望 \(E(\xi)\)。答案应为\(\frac{k}{m}\)的形式，请给出\(k+m\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设正三角形 \(\triangle_{1}\) 的面积为 \(S_{1}\)，作 \(\triangle_{1}\) 的内切圆，再作内切圆的内接正三角形 \(\triangle_{2}\)，面积为 \(S_{2}\)，如此下去作一系列的正三角形 \(\triangle_{3}\), \(\triangle_{4}, \cdots\)，其面积相应为 \(S_{3}, S_{4}, \cdots\)。设 \(S_{1}=1, T_{n} = S_{1} + S_{2} + \cdots + S_{n}\)。求 \(\lim_{n \rightarrow +\infty} T_{n}\) 的值。原始答案的形式是\(\frac{m}{n}\)，请给出m+n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知正三棱柱 \(P-ABC\) 的底面 \(ABC\) 为正三角形，该正三棱柱的外接球的球心 \(O\) 满足 \(\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}=0\)。求二面角 \(A-PB-C\) 的余弦值。答案的形式为 \(\frac{k}{m}\)，请给出 k + m 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设方程
$$
1+x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots-\frac{x^{2018}}{2018}=0
$$
的所有实数根均在区间 $[a, b](a 、 b \in \mathbf{Z}$, $a<b ）$ 内。求 $b-a$ 的最小值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 $a 、 b 、 c$ 满足
$$
\left\{\begin{array}{l}
a^{2}-b c-8 a+7=0 \\
b^{2}+c^{2}+b c-6 a+6=0
\end{array}\right.
$$
求 $a$ 的取值范围。答案的形式为 $[x, y]$，请给出 $x + y$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 $a 、 b 、 c$ 满足 $\left\{\begin{array}{l}a^{2}-b c-8 a+7=0, \\ b^{2}+c^{2}+b c-6 a+6=0,\end{array}\right.$ 求 $a$ 的取值范围的下限和上限。如果取值范围为 $[m, n]$，请给出 $m + n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $M=\left\{(x, y) \mid y \geqslant x^{2}\right\}, N=\left\{(x, y) \mid x^{2}+(y-a)^{2} \leqslant 1\right\}$, 找出使 $M \bigcap N=N$成立的充要条件中$a$的最小值。请将a的最小值乘以100后给出结果。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设复数 $w=\left(\frac{a+\mathrm{i}}{1+\mathrm{i}}\right)^{2}$，其中 $a$ 为实数，若 $w$ 的实部为 2，求 $w$ 的虚部。原始答案为-\frac{k}{m}的格式，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 \(\triangle ABC\) 中, 若三边长 \(a, b, c\) 满足 \(a+c=3b\), 则 \(\tan \frac{A}{2} \tan \frac{C}{2}\) 的值为多少？答案的形式为\(\frac{k}{m}\)，请给出\(k+m\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$A, B$ 是抛物线 $y=3-x^{2}$ 上关于直线 $x+y=0$ 对称的相异两点，求 $|A B|$ 的值。答案的形式为 $k \sqrt{m}$，请给出 $k + m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $\boldsymbol{a} 、 \boldsymbol{b} 、 \boldsymbol{c}$ 为同一平面内的三个单位向量，且 $a \perp b$。求 $(\boldsymbol{c}-\boldsymbol{a}) \cdot(\boldsymbol{c}-\boldsymbol{b})$ 的最大值。答案的形式为 $k + \sqrt{m}$，请给出 $k + m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1, a_{n+1}=n+1+a_{n}\left(n \in \mathbf{N}^{*}\right)$. 若 $[x]$ 表示不超过实数 $x$ 的最大整数,求 $\left[\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{2018}}\right]$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 \(0<x<\frac{\pi}{2}\), 且 \(\frac{\sin ^{4} x}{9}+\frac{\cos ^{4} x}{4}=\frac{1}{13}\), 求 \(\tan x\) 的值。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 \(0<x<\frac{\pi}{2}\), 且 \(\frac{\sin ^{4} x}{9}+\frac{\cos ^{4} x}{4}=\frac{1}{13}\)，求 \(\tan x\) 的值。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 \(x 、 y \in \mathbf{R}\), 且
\(\log _{4}(x+2 y)+\log _{4}(x-2 y)=1\).
求 \(x-|y|\) 的最小值。答案形式为\(\sqrt{k}\)，请给出k的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在边长为 8 的正方形 $A B C D$ 中, $M$ 为边 $B C$ 的中点, $N$ 为边 $D A$ 上一点, 且 $D N=3 N A$. 若对于常数 $m$ ，在正方形 $A B C D$ 的边上恰有六个不同的点 $P$ ，使得 $\overrightarrow{P M} \cdot \overrightarrow{P N}=m$ ，则实数 $m$ 的取值范围是 $($用不等式表示$)$，形如$a < m < b$，请给出$a + b$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知函数 $f(x)=|2 x-4|-3 x+8$ $(x \in \mathbf{R})$ 。求 $f(x)$ 的反函数 $f^{-1}(x)$ 的解析式。答案的形式为\begin{cases}-kx+m, x\leqslant2;\\-\frac{n}{p}x+\frac{q}{r}, x>2\end{cases}，其中p和r不能被5整除。请计算k+m+n+p+q+r的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知二面角 \(\alpha-l-\beta\) 的大小为 \(30^{\circ}\)。求由平面 \(\alpha\) 上的圆在平面 \(\beta\) 上的正射影得到的椭圆的离心率。原始答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知函数 \(f(x)=x^{2}-2 t x+t\)。当 \(x \in [-1,1]\) 时，记 \(f(x)\) 的最小值为 \(m\)。求 \(m\) 的最大值。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设方程
$$
1+x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots-\frac{x^{2018}}{2018}=0
$$
的所有实数根均在区间 $[a, b](a 、b \in \mathbf{Z}, a<b )$ 内. 求 $b-a$ 的最小值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 \( f(x) \) 是定义在实数集上的周期为 2 的函数，且是偶函数。已知当 \( x \in [2,3] \) 时，\( f(x) = x \)。请根据已知条件，推导出当 \( x \in [-2,0] \) 时，\( f(x) \) 的解析式，并将其写成 \( a - |x + b| \) 的形式。请给出 \( a + b \) 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 \( f(x) \) 是定义在实数集上的周期为 2 的函数，且是偶函数。已知当 \( x \in [2,3] \) 时, \( f(x)=x \)。求当 \( x \in [-2,0] \) 时, \( f(x) \) 的解析式。答案的形式为 \( f(x) = a - |x + b| \)，请给出 a + b 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $a$ 是正整数， $a<100$ ，并且 $a^{3}+23$ 能被 24 整除。求满足条件的 $a$ 的个数。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设复平面上单位圆内接正 20 边形的 20 个顶点所对应的复数依次为 $Z_{1}, Z_{2}, \cdots, Z_{20}$, 求复数 $Z_1^{1995}, Z_2^{1995}, \cdots, Z_{20}^{1995}$ 所对应的不同的点的个数。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知方程 $|x-2 n|=k \sqrt{x}(n \in \mathbf{N})$ 在区间 $(2 n-1,2 n+1]$ 上有两个不相等的实根，求 $k$ 的取值范围。答案的形式为 $0<k \leqslant \frac{1}{\sqrt{a n+b}}$，请给出$a+b$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知两个实数集合 $A=\left\{a_{1}, a_{2}, \cdots, a_{100}\right\}$ 与 $B=\left\{b_{1}, b_{2}, \cdots, b_{50}\right\}$, 若从 $A$ 到 $B$ 的映射 $f$ 使得 $B$ 中每个元素都有原象，且 $f\left(a_{1}\right) \leqslant f\left(a_{2}\right) \leqslant \cdots \leqslant f\left(a_{100}\right)$ ，则这样的映射共有多少种？原始答案的形式为$C_{x}^{y}$，请给出x+y的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

删去正整数数列 $1,2,3, \cdots \cdots$ 中的所有完全平方数, 得到一个新数列。求这个数列的第 2003 项。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Two obvious approximations to the length of the perimeter of the ellipse with semi-axes $a$ and $b$ are $\pi(a+b)$ and $2 \pi(a b)^{1/2}$. Which one comes nearer the truth when the ratio $b / a$ is very close to 1? Write your answer in the form of $\pi(x+y)$, where $x$ and $y$ are integers. Find the value of $x + y$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

过抛物线 \(y^{2}=8(x+2)\) 的焦点 \(F\) 作倾斜角为 \(60^{\circ}\) 的直线，若此直线与抛物线交于 \(A 、 B\) 两点，弦 \(AB\) 的中垂线与 \(x\) 轴交于点 \(P\)，则线段 \(PF\) 的长等于多少？答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 \(x, y\) 都在区间 \((-2,2)\) 内，且 \(x y=-1\)，求函数 \(u=\frac{4}{4-x^{2}}+\frac{9}{9-y^{2}}\) 的最小值。答案的形式为\(\frac{k}{m}\)，请给出\(k+m\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在四面体 \(A B C D\) 中，设 \(A B=1\)，\(C D=\sqrt{3}\)，直线 \(A B\) 与 \(C D\) 的距离为 2，夹角为 \(\frac{\pi}{3}\)。求四面体 \(A B C D\) 的体积。答案的形式为\(\frac{a}{b}\)，请给出a+b的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设等差数列的首项及公差均为非负整数，项数不少于 3 ，且各项的和为 $97^{2}$ ，求这样的数列共有多少个。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知数列 $\left\{x_{n}\right\}$ 满足 $x_{n+1}=x_{n}-x_{n-1}(n \geqslant 2), x_{1}=a, x_{2}=b$, 记 $S_{n}=$ $x_{1}+x_{2}+\cdots+x_{n}$. 求$x_{100}$和$S_{100}$的值。答案形式为$kx_{100}, mS_{100}$，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设等差数列的首项及公差均为非负整数，项数不少于 3 ，且各项的和为 $97^{2}$ 。求这样的数列的个数。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $E, F, G$ 分别是正四面体 $A B C D$ 的棱 $A B, B C, C D$ 的中点，求二面角 $C-F G-E$ 的大小。如果答案为 $\pi-\operatorname{arccot} x$，其原始答案的形式为\frac{k\sqrt{n}}{m}，请计算k+m+n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $E 、 F 、 G$ 分别是正四面体 $A B C D$ 的棱 $A B 、 B C 、 C D$ 的中点。求二面角 $C-F G-E$ 的大小，并用表达式 $\pi-\operatorname{arccot} \frac{\sqrt{m}}{n}$ 表示，其中$m$和$n$是整数，且$\frac{\sqrt{m}}{n}$是最简分数。请给出$m + n$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 \(0<x<\frac{\pi}{2}\)，且 \(\frac{\sin ^{4} x}{9}+\frac{\cos ^{4} x}{4}=\frac{1}{13}\)，求 \(\tan x\) 的值。答案形式为 \(\frac{k}{m}\)，请给出 \(k + m\) 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

对于实数 $t$, 已知等比数列 $\left\{a_n\right\}$ 的前三项依次为 $2 t, 5 t-1,6 t+2$, 且该数列的前 $n$ 项和为 $S_n$. 求满足不等式 $|S_n-\frac{1}{65}|<1$ 的最大整数 $n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 \(0<k<4\)，直线 \(l_1: k x-2 y-2 k+8=0\) 和直线 \(l_2: 2 x+k^2 y-4 k^2-4=0\) 与两坐标轴围成一个四边形。求使这个四边形面积最小的 \(k\) 值。答案的形式为\(\frac{m}{n}\)，请给出\(m+n\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

对于实数 $t$, 已知等比数列 $\left\{a_{n}\right\}$ 的前三项依次为 $2 t, 5 t-1,6 t+2$, 且该数列的前 $n$ 项和为 $S_{n}$. 求满足不等式 $\left|S_{n}-\frac{1}{65}\right|<1$ 的最大整数 $n$ 的值。设答案为$m$，请给出$m$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

长度分别为 $1, a, a, a, a, a$ 的线段能成为同一个四面体的 6 条棱的充要条件是 $a$ 满足的条件是 \frac{\sqrt{k}}{m}，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

记 \(F(x, y) = (x-y)^2 + \left( \frac{x}{3} + \frac{3}{y} \right)^2 \quad (y \neq 0)\)。求 \(F(x, y)\) 的最小值。原始答案的形式为\(\frac{k}{m}\)，请计算\(k + m\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

On the same side of a straight line, three circles are drawn as follows: a circle with a radius of $4$ inches is tangent to the line. The other two circles are equal, and each is tangent to the line and to the other two circles. What is the radius of the equal circles in inches?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form. Find the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求$(x-1)(x-2)(x-3)(x-4)$的最小值，其中$x$为实数。请给出最小值的整数部分。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Equilateral triangle \(ABC\) has been creased and folded so that vertex \(A\) now rests at \(A'\) on \(\overline{BC}\) as shown. If \(BA' = 1\) and \(A'C = 2\), then find the length of the crease \(\overline{PQ}\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\), please give the value of k + m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given a right triangle $ABC$ with legs $BC=3$ and $AC=4$, find the length of the shorter angle trisector from $C$ to the hypotenuse. Provide your answer in the form \(\frac{k\sqrt{m}-n}{p}\), and then find the value of k + m + n + p.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the maximum value of the function \(\frac{1}{|x+1| + |x+2| + |x-3|}?\) The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A palindrome, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x + 32$ are three-digit and four-digit palindromes, respectively. Find the sum of the digits of $x$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Which value of $x$ satisfies the equation $\sqrt[3]{6+\sqrt{x}} + \sqrt[3]{6-\sqrt{x}} = \sqrt[3]{3}$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Consider two concentric circles with radii \(17\) and \(19\). The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle? The original answer is in the form \(k\sqrt{m}\). Please find the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Three digits are selected at random and without replacement from the set of digits $0$ through $9$. Find the probability that the three digits can be arranged to form a multiple of $5$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If \(b > 1\), \(x > 0\) and \((2x)^{\log_b 2} - (3x)^{\log_b 3} = 0\), find the value of \(x\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many real pairs $(x, y)$ are there such that
\[
x^2 + 2y = 2xy \\
x^3 + x^2y = y^2
\]
Find the number of real pairs $(x, y)$ that satisfy the equations.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many triples of positive integers $(a,b,c)$ are there such that $a!+b^3 = 18+c^3$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the total number of solutions of the equation $xyz=2520$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If $ \log 2 = 0.3010$ and $ \log 3 = 0.4771$, find the value of $ x$ when $ 3^{x+3} = 135$. If the answer is $y$, what is the value of $100y$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the sum \( \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \cdots + \frac{1}{1996 \cdot 1997 \cdot 1998} \)? Provide your answer in the form of \(\frac{k}{m} - \frac{n}{p \cdot q \cdot r}\). Please find the value of \(k + m + n + p + q + r\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a > b > 0$. The area of $PQRS$ is $16$. What is the value of $a+b$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a$, $b$, $c$, $d$, and $e$ be distinct integers such that
\[ (6 - a)(6 - b)(6 - c)(6 - d)(6 - e) = 45. \]
Find the value of $a + b + c + d + e$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Andrea inscribed a circle inside a regular pentagon and circumscribed a circle around it, calculating the area of the region between the two circles. Bethany performed the same operations with a regular heptagon (7 sides). The areas of the two regions were $A$ and $B$, respectively. Each polygon had a side length of $2$. The relationship between $A$ and $B$ can be expressed in the form $A = kB$. Please find the value of $k$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Two integers have a sum of $26$. When two more integers are added to the first two integers, the sum is $41$. Finally, when two more integers are added to the sum of the previous four integers, the sum is $57$. Find the minimum number of even integers among the $6$ integers.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A circular grass plot with a diameter of 12 feet is intersected by a straight gravel path that is 3 feet wide. One edge of the path passes through the center of the plot. Calculate the number of square feet in the remaining grass area. The original answer is in the form $k\pi - m\sqrt{n}$, please provide the value of k + m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$(a_n)_{n = 0}^\infty$ is a sequence of integers. For every $n \ge 0$, $a_{n + 1} = a_n^3 + a_n^2$. Find the maximum number of distinct residues of $a_i$ in $\pmod{11}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, find the largest possible difference between the two primes.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last two marbles. Find the probability that Cheryl gets $2$ marbles of the same color. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

At each step, we change the positions of exactly two numbers from the sequence $1, 2, 3, 4, 5, 6, 7$. How many different arrangements can be formed after two steps?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. Determine how many values of $n$ satisfy the equation \( n + S(n) + S(S(n)) = 2007 \).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the maximum area of an isosceles trapezoid that has legs of length \(1\) and one base twice as long as the other? The maximum area can be expressed in the form \(\frac{k}{m}\), where \(k\) and \(m\) are coprime integers. Please find the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In $\triangle ABC$, $AB = 8$, $BC = 7$, $CA = 6$, and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Find the length of $PC$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $S$ be the set of all those 2007-place decimal integers $\overline{2s_1a_2a_3 \ldots a_{2006}}$ which contain an odd number of digit $9$ in each sequence $a_1, a_2, a_3, \ldots, a_{2006}$. The cardinal number of $S$ is given in the form \frac{1}{x}(y^{2006} - z^{2006}). Please calculate the value of x + y + z.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the equation \(\frac{x(x - 1) - (m + 1)}{(x - 1)(m - 1)} = \frac{x}{m}\), find the value of \(m\) such that the roots are equal. The original answer is in the form \(-\frac{a}{b}\). Please provide the value of \(a + b\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A particle of mass $m$ moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$.

The fractional energy transfer is the absolute value of the final kinetic energy of $M$ divided by the initial kinetic energy of $m$.

If the collision is perfectly elastic, under what condition will the fractional energy transfer between the two objects be a maximum? Express your answer as a relationship between $m$ and $M$ in the form k*m = n*M, and provide the sum of k and n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A vertical mass-spring oscillator is displaced $2.0\, \text{cm}$ from equilibrium. The $100\, \text{g}$ mass passes through the equilibrium point with a speed of $0.75\, \text{m/s}$. What is the spring constant of the spring? If the spring constant is $k\, \text{N/m}$, what is the value of $k$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If one can find a student with at least $k$ friends in any class which has 21 students such that at least two of any three of these students are friends, what is the largest possible value of $k$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If $ABCD$ is a $2 \times 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, find the area of quadrilateral $BEIH$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设\(\boldsymbol{a}、\boldsymbol{b}、\boldsymbol{c}\)为同一平面内的三个单位向量，且 \(\boldsymbol{a}\perp\boldsymbol{b}\)。求\((\boldsymbol{c}-\boldsymbol{a})\cdot(\boldsymbol{c}-\boldsymbol{b})\)的最大值。答案的形式为\(m + \sqrt{n}\)，请给出\(m+n\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

如图 1, 设 $P$ 为 $\triangle A B C$ 内一点, 且 $\boldsymbol{A P}$ $=\frac{2}{5} \boldsymbol{A B}+\frac{1}{5} \boldsymbol{A} \boldsymbol{C}$ 。求 $\frac{S_{\triangle A B P}}{S_{\triangle A B C}}$ 的值。答案应为\frac{k}{m}的形式，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

如图, 在棱长为 1 的正四面体 $A B C D$ 中, $G$ 为 $\triangle B C D$ 的重心, $M$ 是线段 $A G$ 的中点, 求三棱锥 $M-B C D$ 的外接球的表面积除以 $\pi$ 的值。答案的形式为\(\frac{k}{m}\)，请给出\(k+m\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 \(x, y\in\mathbf{R}\), 且\(\log _{4}(x+2 y)+\log _{4}(x-2y)=1\), 求\(x-|y|\)的最小值。原始答案是以\(\sqrt{k}\)的形式给出，请给出k的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设函数$f(x)=x^{3}+a x^{2}+b x+c$（$a, b, c$均为非零整数），若 $f(a)=a^{3}, f(b)=b^{3}$，则 $c$ 的值是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在凸四边形 $A B C D$ 中, 若 $B C=4, \angle A D C=60^{\circ}, \angle B A D=90^{\circ}$, 四边形 $A B C D$ 的面积等于 $\frac{A B \cdot C D+B C \cdot A D}{2}$, 则 $C D$ 的长乘以10后的值（精确到小数点后1位）是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

四个半径为 1 的球两两相切，则它们外切正四面体的棱长为多少？答案的形式为 $k(1+\sqrt{m})$，请给出 $k+m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $k \neq 1$ ，则等比数列 $a+\log _{2} k, a+\log _{4} k, a+\log _{8} k$ 的公比是多少？答案的形式为\frac{k}{m}，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$f(x)$ 是定义在实数集 $\mathbf{R}$ 上的函数，满足 $\forall x \in \mathbf{R}, 2 f(x)+f\left(x^{2}-1\right)=1$ ，求 $f(-\sqrt{2})$ 的值。答案的形式为 \(\frac{k}{m}\)，请给出\(k+m\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知正方形 $A B C D$ 的边长为 1, 若 $P_{1}, P_{2}, P_{3}, P_{4}$ 是正方形内部的 4 个点，使得 $\triangle A B P_{1}, \triangle B C P_{2}, \triangle C D P_{3}$ 和 $\triangle D A P_{4}$ 都是正三角形，则四边形 $P_{1} P_{2} P_{3} P_{4}$ 的面积等于多少？答案的形式为k-\sqrt{m}，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

使得 $p^{3}+7 p^{2}$ 为完全平方数的不大于 100 的素数 $p$ 的个数为多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

函数 $f(x)=x(x+1)(x+2)(x+3)$ 的最小值是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

不等式组 \(\left\{\begin{array}{l}y \geqslant 2|x|-1 \\ y \leqslant-3|x|+5\end{array}\right.\) 所表示的平面区域的面积是多少？答案应为\(\frac{k}{m}\)的形式，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $O$ 是凸四边形 $A B C D$ 对角线 $A C$ 和 $B D$ 的交点, $\triangle A O B, \triangle B O C$, $\triangle C O D, \triangle D O A$ 的周长相等. 若 $\triangle A O B, \triangle B O C, \triangle C O D$ 的内切圆半径分别为 $3,4,6$, 则 $\triangle D O A$ 的内切圆半径的形式为\(\frac{k}{m}\)，请给出k+m的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若某三角形的三边长为三个连续正整数，且该三角形有一个角是另一个角的 2 倍，请找出这个三角形的三边长。答案的格式为(k, m, n)，请给出k + m + n的值。

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