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 $f(t)=t^{2}+2t$ ，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.

若数列 \(\left\{a_{n}\right\}\) 的前 \(n\) 项和 \(S_{n}=n^{2} a_{n}\)，且 \(a_{1}=2019\)，求 \(a_{2018}\) 的值。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.

若正数 \(a, b\) 满足 \(a+b=1\)，则求 \(\frac{1}{a}+\frac{27}{b^{3}}\) 的最小值。答案的形式为\(\frac{k+m \sqrt{3}}{n}\)，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.

设实数 $x ,y$ 满足 $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$, 求 $\sqrt{x^{2}+y^{2}-2 y+1}+\sqrt{x^{2}+y^{2}-2 x+1}$ 的最小值。答案的形式为 $k\sqrt{m} - \sqrt{n}$，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.

15 人围坐在圆桌旁，选出 4 人使得其中任意两人都不相邻的选法数为多少？

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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.

设集合 $S$ 中有 10 个元素, 从 $S$ 中每次随机选取 1 个元素, 取出后还放回 $S$ 中, 则取5次后出现重复元素的概率是（保留两位有效数字）。如果答案是 $x$，那么请给出 $100x$ 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.

一个正整数 $n$ 称为具有 3 -因数积性质。若 $n$ 的所有正因数的乘积等于 $n^{3}$ 。求不超过 400 的正整数中具有 3 -因数积性质的数的个数。

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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 \(A\) 是不超过 2018 的正整数组成的集合，对于正整数 \(k\)，用 \(a_{k}\) 表示所有可能的 \(A\)中 \(k\) 个数乘积的倒数之和。求 \(a_{2}+a_{4}+\cdots+a_{2018}\) 的值。答案应以\(\frac{k}{m}\)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.

用 $[x]$ 表示不超过实数 $x$ 的最大整数。例如 $[\pi]=3$, $[-\pi]=-4$ 。设 $n$ 为正整数。用 $a_{n}$ 表示当 $x \in[0, n)$ 时, 函数 $f(x)=[x[x]]$ 的值域中的元素的个数。求使得 $\frac{a_{n}+2018}{n}$ 最小的 $n$ 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 \(\triangle A B C\) 的面积为 1, \(D, E\) 分别为边 \(BC, CA\) 上的点, 且 \(B D=\frac{1}{3} B C\), \(C E=\frac{1}{3} C A\), \(AD\) 和 \(BE\) 交于点 \(P\)。求四边形 \(PDCE\) 的面积。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.

$1 \times 1!+2 \times 2!+\cdots+672 \times 672!$ 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.

若存在正整数 $k<n\left(n \geqslant 3, n \in \mathrm{~N}^{*}\right)$, 使得 $\mathrm{C}_{n}^{k-1}, \mathrm{C}_{n}^{k}, \mathrm{C}_{n}^{k+1}$ 构成等差数列，其中组合数 $\mathrm{C}_{n}^{k}=$ $\frac{n!}{k!(n-k)!}(n!=1 \cdot 2 \cdot 3 \cdot \cdots \cdot n$, 约定 $0!=1)$ ，则称 $n$ 为理想数。不超过 2020 的理想数的个数是多少？

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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（2019 $\cdot$ 2020）2021 的全体正因数中选出若干个，使得其中任意两个之积都不是完全平方数，则最多可选的正因数个数为多少？

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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.

若数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1, a_{2}=4$, 且 $a_{n}^{2}-$ $a_{n-1} a_{n+1}=2^{n-1}\left(n \geqslant 2, n \in \mathbf{N}^{*}\right)$ ，求 $a_{2020}$ 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.

若 $a, b, c, d$ 是方程 $x^{4}+2 x^{3}+3 x^{2}+4 x+5= 0$ 的全部复数根, 求 $\frac{a-1}{a+2}+\frac{b-1}{b+2}+\frac{c-1}{c+2}+\frac{d-1}{d+2}$ 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.

已知等边 \(\triangle ABC\) 的边长为1，过点 \(C\) 作以边 \(AB\) 为直径的圆的切线交线段 \(AB\) 的延长线于点 \(D\)，且 \(AD > BD\)，求 \(\triangle BCD\) 的面积。原始答案的形式为 \(\frac{k \sqrt{2} - m \sqrt{3}}{n}\)，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.

方程 $19 x+93 y=4 x y$ 的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.

从圆 \(x^{2}+y^{2}=4\) 上的点向椭圆 \(C: \frac{x^{2}}{2}+y^{2}= 1\) 引切线，两个切点间的线段称为切点弦。求所有的切点弦所在的直线围成区域的面积。原始答案的形式为\(\frac{k\pi}{m}\)，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 \(P A B C\) 中，\(A B = P A = 2 A C = 2\)，且 \(P A \perp\) 平面 \(A B C\)，\(\angle B A C = 120^{\circ}\)。求该四面体的外接球的表面积。如果答案是 \(\frac{x \pi}{3}\)，求出 \(x\) 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.

已知 P 为单位圆上一动点, 且 $A(0,2), B(0,-1)$, 求 $|A P| \cdot|B P|^{2}$ 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.

在三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, 已知 $\angle A B C=90^{\circ}, A B=6, B C=B B_{1}=3 \sqrt{2}$, 动点 P 在线段 $B_{1} C$ 上，求 $A_{1} P+B P$ 的最小值。答案形式为 $k\sqrt{m}$，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.

令 $z_{k}=\cos \frac{2 k-2}{5} \pi+i \sin \frac{2 k-2}{5} \pi(k=1,2,3,4,5)$ ，若 $a_{i}=\prod_{i \neq j}\left(z_{i}-z_{j}\right)$ ，请计算 $a_{1} a_{2} a_{3} a_{4} a_{5}$ 的值，并将结果表示为一个整数。

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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.

A truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force $F$ to the truck. The truck rolls a distance $x$ before coming to a stop, and the time it takes to stop is $t$.

Find the expression that represents the initial kinetic energy of the truck (i.e., the kinetic energy before the driver starts braking). The answer should be in the form of $kF \cdot mx$, where $k$ and $m$ are 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.

Let \(f(x)\) be a real-valued function satisfying \(af(x) + bf(-x) = px^2 + qx + r\). Here, \(a\) and \(b\) are distinct real numbers, and \(p\), \(q\), \(r\) are non-zero real numbers. Determine the condition under which \(f(x) = 0\) will have real solutions. The original answer is in the form \(\left(\frac{k(a+b)}{m(a-b)}\right) \leq \frac{q^2}{4pr}\). 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.

设复数 $z$ 满足 $|3 z-7 i|=3$ ，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 \(\alpha, \beta\) 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和m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.

设多项式 \(f(x)\) 的各项系数都是非负实数，且 \(f(1)=f^{\prime}(1)=f^{\prime \prime}(1)=f^{\prime \prime \prime}(1)=1\)。求 \(f(x)\) 的常数项的最小值。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.

设定点 $A, B, C, D$ 是以点 $O$ 为中心的正四面体的顶点，用 $\sigma$ 表示空间以直线 $O A$ 为轴且满足条件 $\sigma(B)=C$ 的旋转，用 $\tau$ 表示空间关于点 $O, C, D$ 所在平面的镜面反射，设 $l$ 为过 $A B$ 的中点与 $C D$ 的中点的直线，用 $\omega$ 表示空间以 $l$ 为轴的 $180^{\circ}$ 旋转。设 $\sigma \circ \tau$ 表示变换的复合，先作 $\tau$ ，再作 $\sigma$ ，则 $\omega$ 可以表示为 $k \circ m \circ k \circ k \circ m \circ k$。请计算 $k$ 和 $m$ 出现的次数之和。

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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.

若 $A+B=\frac{2 \pi}{3}$, 求 $\cos ^{2} A+\cos ^{2} B$ 的最小值和最大值，分别记为 $x$ 和 $y$。答案的形式分别为x = \frac{m}{n}和y = \frac{p}{q}，请给出m+n+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.

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？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.

有 $n$ 个质点, 每个质点质量为 $m_{k}$, 则质心位置 $x=\frac{\sum m_{k} x_{k}}{\sum m_{k}}$。对于一杆，长 3 m，放于 $x \in[-1,2]$ 间，且线密度满足 $\beta=2+x$，求质心的位置。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 $[x]$ 为高斯函数，求满足 $\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{5}\right]=x$ 的解的组数。

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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.

设复数 \(z\) 满足 \(|3z - 7i| = 3\)，求 \(\left|\frac{z^2 - 2z + 2}{z - 1 + i}\right|\) 的最大值和最小值。如果最大值为 \(\frac{m}{n}\)，最小值为 \(\frac{p}{q}\)，请给出 \(m + n + p + q\) 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.

若复数 $z$ 满足 $\left|\frac{1}{z}+z\right|=1$, 则求 $|z|$ 的两个可能的值。答案的形式分别为 \frac{\sqrt{k}-m}{n} 和 \frac{\sqrt{p}+q}{r}，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.

由 \(y \geqslant 2|x|-1\) 和 \(y \leqslant-3|x|+5\) 所围成的平面区域的面积为多少？请注意，答案的形式为\(\frac{k}{m}\)，其中k和m是互质的正整数。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.

已知等腰梯形 ABCD 中，AD\parallel BC，BC=2AD=4，\angle ABC=60^\circ，\overrightarrow{CE}=\frac{1}{3}\overrightarrow{CD}，求 \overrightarrow{CA}\cdot\overrightarrow{BE} 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.

在正四面体 ABCD 中，棱 AD 的中点和面 BCD 的中心的连线为 MN，棱 CD 的中点和面 ABC 的中心的连线为 PQ。求 MN 与 PQ 所成角的余弦值。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.

已知函数 \( f(x) = \sqrt{2x^4-18x^2+12x+68}+x^2-x+1 \)，求 \( f(x) \) 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.

已知 \(A\), \(B\), \(C\) 是平面上三个不同点，且 \(BC = a\), \(CA = b\), \(AB = c\)，求 \(\frac{c}{a+b}+\frac{b}{c}\) 的最小值。答案应为\(\sqrt{k} - \frac{m}{n}\)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.

双曲线 \(x^2-y^2=2022\) 上格点（横纵坐标均为整数的点）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.

What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation?

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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.

Suppose $a$ and $b$ are digits which are not both nine and not both zero. If the repeating decimal $0.\overline{AB}$ is expressed as a fraction in lowest terms, find how many possible denominators there are.

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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 $0 \le x_0 <1$, let \[ x_n= \begin{cases} 2x_{n-1} & \text{if}\ 2x_{n-1} <1 \\ 2x_{n-1}-1 & \text{if}\ 2x_{n-1} \ge 1 \end{cases} \] for all integers $n>0$. Find the number of values of $x_0$ for which $x_0=x_5$.

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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 adjoining figure, $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, find the area of $CMN$ in square inches. The original answer is in the form \(k\sqrt{n} - m\), please give the value of k + n + 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 bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to \((1,0)\). Then it makes a \(90^\circ\) counterclockwise turn and travels \(\frac{1}{2}\) a unit to \(\left(1, \frac{1}{2} \right)\). If it continues in this fashion, each time making a \(90^\circ\) degree turn counterclockwise and traveling half as far as the previous move, to which point \((x, y)\) will it come closest? Provide the answer in the form of two fractions \(\left(\frac{a}{b}, \frac{c}{d}\right)\), and calculate the value of a + b + c + d.

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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 $f(x) = x^2 + 6x + 1$, and let $R$ denote the set of points $(x,y)$ in the coordinate plane such that
\[ f(x) + f(y) \leq 0 \text{ and } f(x) - f(y) \leq 0 \]
Find the integer that is closest to the area of $R$.

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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.

Two circles of radius 1 are to be constructed as follows. The center of circle \(A\) is chosen uniformly and at random from the line segment joining \((0,0)\) and \((2,0)\). The center of circle \(B\) is chosen uniformly and at random, and independently of the first choice, from the line segment joining \((0,1)\) to \((2,1)\). Find the probability that circles \(A\) and \(B\) intersect. The original answer is in the form \(\frac{k \sqrt{m} - n}{p}\), please give 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.

A convex polyhedron $Q$ has vertices $V_1, V_2, \ldots, V_n$, and 100 edges. The polyhedron is cut by planes $P_1, P_2, \ldots, P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. Find the number of edges of the polyhedron $R$.

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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.

In triangle $ABC$, lines $CE$ and $AD$ are drawn such that \( \frac{CD}{DB} = \frac{3}{1} \) and \( \frac{AE}{EB} = \frac{3}{2} \). Let \( r = \frac{CP}{PE} \), where $P$ is the intersection point of $CE$ and $AD$. Find the value of \( r \).

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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.

How many ordered pairs of integers $(x,y)$ are there such that $2011y^2 = 2010x + 3$?

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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.

In base $R_1$, the expanded fraction $F_1$ becomes $0.373737\ldots$, and the expanded fraction $F_2$ becomes $0.737373\ldots$. In base $R_2$, fraction $F_1$, when expanded, becomes $0.252525\ldots$, while fraction $F_2$ becomes $0.525252\ldots$. Find the sum of $R_1$ and $R_2$, each written in base ten.

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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.

Fermat determines that since his final exam counts as two tests, he only needs to score a $28$ on it for his test average to be $70$. If he gets a perfect $100$ on the final exam, his average will be $88$. What is the lowest score Fermat can receive on his final and still have an average of $80$?

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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 $ABC$, $AB = AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP = PC = CB$, then if the answer is $x^\circ$, what is the value 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.

Find the last 5 digits of the number $M = 5^{2010}$.

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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.

Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and randomly presses up or down at each floor. After doing this $482$ times, the probability that Dave is on the first floor can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is the remainder when $m+n$ is divided by $1000$?

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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 object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? Write 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.

The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a negligible height above the equator, find the best estimate of the number of hours of flight.

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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 circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. If the measure of the angle subtended at the center of the circle by the shortest side is $x^\circ$, what is the value 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.

A point \(P\) lies in the same plane as a given square with side length \(1\). The vertices of the square, taken counterclockwise, are labeled as \(A\), \(B\), \(C\), and \(D\). The distances from point \(P\) to the vertices \(A\), \(B\), and \(C\) are \(u\), \(v\), and \(w\), respectively. Determine the greatest distance that \(P\) can be from vertex \(D\), given that \(u^2 + v^2 = w^2\). The original answer is in the form \(k + \sqrt{m}\). 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.

At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, find the number of campers who are in exactly two classes.

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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 Scott rolls four fair six-sided dice, find the probability that he rolls more 2's than 1's. The probability is given in the form \(\frac{m}{n}\). Please calculate the value of 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.

If \(\frac{x}{x-1} = \frac{y^2+2y-1}{y^2-2y-2}\), find the value of \(x\). The original answer can be expressed in the form \(ay^2 + by + c\). Please 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.

If \( |x - \log y| = x + \log y \) where \( x \) and \( \log y \) are real, find the expression that represents the relationship between \( x \) and \( y \). The original answer is in the form \( mx(y-n)=p \). Please provide 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.

Find the first step in finding the product \((3x + 2)(x - 5)\) by use of the distributive property in the form \(a(b + c) = ab + ac\). The original answer is in the form \((a+b)x + (a+b)(c)\). Please give 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.

The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)

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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 \(\triangle ABC\), point \(D\) is on \(AC\) and point \(F\) is on \(BC\). It is given that \(AB \perp AC\), \(AF \perp BC\), and \(BD = DC = FC = 1\). Find the length of \(AC\). The original answer is in \(\sqrt[k]{m}\) 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.

For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3 + ax^2 + bx + 6$ have 3 distinct integer roots?

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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.

Halfway through a $100$-shot archery tournament, Chelsea leads by $50$ points. For each shot, a bullseye scores $10$ points, with other possible scores being $8$, $4$, $2$, and $0$ points. Chelsea always scores at least $4$ points on each shot. If Chelsea's next $n$ shots are bullseyes, she will be guaranteed victory. What is the minimum value for $n$?

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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.

Equilateral $\triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $T$ and tangent to sides $AB$ and $AC$ at points $P$ and $Q$. If side $BC$ has length $12$, find the length of segment $PQ$.

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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.

From a two-digit number $N$, we subtract the number with the digits reversed and find that the result is a positive perfect cube. How many values of $N$ satisfy this condition?

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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 $\frac{8}{19}$ of the product of the largest two elements of a positive integer set is not greater than the sum of the other elements, what is the minimum possible value of the largest number in the set?

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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 \(ABCD\) be a regular tetrahedron and let \(E\) be a point inside the face \(ABC\). Denote by \(s\) the sum of the distances from \(E\) to the faces \(DAB\), \(DBC\), \(DCA\), and by \(S\) the sum of the distances from \(E\) to the edges \(AB\), \(BC\), \(CA\). The original answer is in \(\frac{k\sqrt{m}}{n}\) format. 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.

Circles with centers \(P\), \(Q\), and \(R\), having radii \(1\), \(2\), and \(3\), respectively, lie on the same side of line \(l\) and are tangent to \(l\) at \(P'\), \(Q'\), and \(R'\), respectively, with \(Q'\) between \(P'\) and \(R'\). The circle with center \(Q\) is externally tangent to each of the other two circles. Find the area of triangle \(PQR\). The original answer is in the form \(\sqrt{a}-\sqrt{b}\), 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 16-step path is to go from $( -4, -4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. Find the number of such paths that stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step.

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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 solutions does the equation $\tan{(2x)} = \cos{\left(\frac{x}{2}\right)}$ have on the interval $[0, 2\pi]$? Write 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.

A rectangle is inscribed in a triangle such that its base coincides with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, find the expression for $x$ in terms of $b$ and $h$. The original answer is in the form \frac{kb}{mh+b}, 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 \(x \in \left[-\frac{5\pi}{12}, -\frac{\pi}{3}\right]\), find the maximum value of \(y = \tan\left(x + \frac{2\pi}{3}\right) - \tan\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right)\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\). 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.

If $n$ is any whole number, find the largest number by which $n^2(n^2 - 1)$ is always divisible.

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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 every \( x, y \in \mathbb{R}^+ \), the function \( f: \mathbb{R}^+ \to \mathbb{R} \) satisfies the condition \( f(x) + f(y) = f(x)f(y) + 1 - \frac{1}{xy} \). If \( f(2) < 1 \), find the value of \( f(3) \). 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.

For each $x$ in $[0,1]$, define
\[ f(x) = \begin{cases} 2x, & \text{if } 0 \leq x \leq \frac{1}{2}; \\
2 - 2x, & \text{if } \frac{1}{2} < x \leq 1. \end{cases} \]
Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. Find the number of values of $x$ in $[0,1]$ for which $f^{[2005]}(x) = \frac{1}{2}$. The original answer is in the form of $k^m$, 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.

In this diagram, $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which an equilateral triangle $DEF$ is inscribed. Designate angle $BFD$ by $a$, angle $ADE$ by $b$, and angle $FEC$ by $c$. The relationship between $a$, $b$, and $c$ is such that $k(b + c) = 2a$. Please find the value of $k + m + n$, where $m$ and $n$ are the coefficients of $b$ and $c$, respectively.

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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 a standard die, one of the dots is removed at random, with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? 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.

A planet orbits around a star $S$, as shown in the figure. The semi-major axis of the orbit is $a$. The perigee, namely the shortest distance between the planet and the star, is $0.5a$. When the planet passes point $P$ (on the line through the star and perpendicular to the major axis), its speed is $v_1$. What is the value of $v_2$ in terms of $v_1$ when it passes the perigee? The original answer is in the form \(\frac{k}{\sqrt{m}}v_1\). Please 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 \(\triangle ABC\), \(\angle C = 90^\circ\), \(AC = 6\), and \(BC = 8\). Points \(D\) and \(E\) are on \(\overline{AB}\) and \(\overline{BC}\), respectively, and \(\angle BED = 90^\circ\). If \(DE = 4\), find the value of \(BD\). 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.

In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$, and $(5,0)$. Find the slope of the line through the origin that divides the area of this region exactly in half. The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is a simplified fraction. 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 ordered integer pairs $(x,y)$ are there such that $xy = 4(y^2 + x)$?

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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.

A positive integer $n$ has 60 divisors and $7n$ has 80 divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?

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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.

Lily has an unfair coin that has a \(\frac{2}{3}\) probability of showing heads and a \(\frac{1}{3}\) probability of showing tails. She flips the coin twice. What is the probability that the first flip is heads while the second is tails? Provide 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.

Jamal wants to store $30$ computer files on floppy disks, each of which has a capacity of $1.44$ megabytes (MB). Three of his files require $0.8$ MB of memory each, $12$ more require $0.7$ MB each, and the remaining $15$ require $0.4$ MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?

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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 base of a triangle is $15$ inches. Two lines are drawn parallel to the base, terminating in the other two sides, and dividing the triangle into three equal areas. Find the length of the parallel closer to the base in inches. The answer should be in the form $k\sqrt{m}$, where k and m are integers. Please 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.

On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was $15$ inches. If the answer is $x$ inches, what is 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.

For all integers $n$ greater than $1$, define $a_n = \frac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Find the value of $b - c$.

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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.

A ball of radius \(R\) and mass \(m\) is magically put inside a thin shell of the same mass and radius \(2R\). The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface? 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 subset of the integers $1, 2, \ldots, 100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?

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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.

$N$ is a $50$-digit number (in the decimal scale). All digits except the $26^{\text{th}}$ digit (from the left) are $1$. If $N$ is divisible by $13$, what is the value of the $26^{\text{th}}$ digit?

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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 rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $(a, b)$?

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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.

Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. The length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure 2, the length $s$ is found to be $28$ inches. How high is the table 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.

The graphs of $2x + 3y - 6 = 0$, $4x - 3y - 6 = 0$, $x = 2$, and $y = \frac{2}{3}$ intersect in how many points?

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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 $p$ and $q$ be two consecutive terms of the sequence of odd primes. Find the minimum number of positive divisors of $p + q$.

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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.

What are the last three digits of $49^{303} \cdot 3993^{202} \cdot 39^{606}$?

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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 following base-10 equation, each letter represents a unique digit: $AM \cdot PM = ZZZ$. Find the sum of $A + M + P + Z$.

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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.

What is the product of all odd positive integers less than 10000? Express your answer in the form of a factorial expression divided by powers of 2 and factorials. The original answer is in the format \(\frac{x!}{2^y \cdot z!}\). Please find 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.

What is the area of the region enclosed by the graph of the equations $x^2 - 14x + 3y + 70 = 21 + 11y - y^2$ that lies below the line $y = x - 3$? If the area is $A\pi$, what is the value of $A$?

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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 point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position? If the new position is $(x, y)$, please give the coordinates in the form of $(a, b)$ and 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.

My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, find the expected score of the losing team. 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.

Two circles are externally tangent. Lines \(\overline{PAB}\) and \(\overline{PA'B'}\) are common tangents with \(A\) and \(A'\) on the smaller circle and \(B\) and \(B'\) on the larger circle. If \(PA = AB = 4\), then what is the value of \(x\) if the area of the smaller circle is \(x\pi\)?

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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.

$AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $\angle BOC$ is $60^\circ$. If the diameter of the circle is $5$ inches, find the length of chord $AC$, expressed in inches. The answer should be in the form \(\frac{k\sqrt{m}}{n}\), where \(k\), \(m\), and \(n\) are integers. Please calculate 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.

For how many integers $n$, with $2 \leq n \leq 80$, is $\frac{(n-1)n(n+1)}{8}$ equal to an integer? Provide the number of such 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.

Arithmetic sequences $(a_n)$ and $(b_n)$ have integer terms with $a_1 = b_1 = 1 < a_2 \le b_2$ and $a_nb_n = 2010$ for some $n$. What is the largest possible value of $n$?

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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.

Let $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Let $D$, $E$, $F$ be points on sides $[AB]$, $[AC]$, $[BC]$, respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Let $H_1$, $H_2$, $H_3$ be the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$, respectively. Find the area of triangle $H_1H_2H_3$.

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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 sum of distinct residues of the number $2012^n + m^2 \pmod{11}$ where $m$ and $n$ are positive 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.

The sides of a triangle have lengths $11$, $15$, and $k$, where $k$ is an integer. Find the number of integer values of $k$ for which the triangle is obtuse.

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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 unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex \(W\)? 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.

Given $a_1 = 1$, $a_{n+1} = \frac{a_n}{\sqrt{1 + 4a_n^2}}$ for $n \ge 1$, find the least $k$ such that $a_k < 10^{-2}$.

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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 solutions does the equation $\sin \left( \frac{\pi}{2} \cos x\right)=\cos \left( \frac{\pi}{2} \sin x\right)$ have in the closed interval $[0,\pi]$? 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.

How many different solutions does the congruence $x^3 + 3x^2 + x + 3 \equiv 0 \pmod{25}$ have? 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.

Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus, the numbers $29$, $23$, $21$, and $17$ are obtained. Find one of the original 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 simplified model of a bicycle of mass \(M\) has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is \(w\), and the center of mass of the bicycle is located midway between the tires and a height \(h\) above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude \(a\). Air resistance may be ignored.

Case 1: Assume that the coefficient of sliding friction between each tire and the ground is \(\mu\), and that both tires are skidding: sliding without rotating. Express your answers in terms of \(w\), \(h\), \(M\), and \(g\).

What is the maximum value of \(a\) so that both tires remain in contact with the ground? The original answer is in the form \(\frac{kwg}{mh}\), 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.

Let the bisectors of the exterior angles at \(B\) and \(C\) of triangle \(ABC\) meet at \(D\). Then, if all measurements are in degrees, find the value of angle \(BDC\). The original answer is in the form \(\frac{k}{m}(n - A)\). 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.

At most, how many elements can a set have such that all elements are less than $102$ and the set does not contain the sum of any two of its elements? 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.

In a bag of marbles, \(\frac{3}{5}\) of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? The original answer is in the form \(\frac{k}{m}\), where the fraction is in simplest form. 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.

Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.

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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 $\overline{AB}$ be a diameter in a circle of radius $5\sqrt{2}$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt{5}$ and $\angle AEC = 45^{\circ}$. Find the value of $CE^2+DE^2$.

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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 sum of the $2$ smallest prime factors of $2^{1024} - 1$.

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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 least positive integer $k$ such that $n+k \in S$ for every $n \in S$, where $S = \{ n : n3^n + (2n+1)5^n \equiv 0 \pmod{7} \}$?

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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 $x_1=5, x_2=401$, and \[ x_n=x_{n-2}-\frac{1}{x_{n-1}} \] for every $3\leq n \leq m$, find the largest value of $m$.

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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.

Calculate the product \(\frac{1}{3} \times \frac{2}{4} \times \frac{3}{5} \times \cdots \times \frac{18}{20} \times \frac{19}{21}\). What is the value of this product in its simplest form? 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.

The parallelogram bounded by the lines $y = ax + c$, $y = ax + d$, $y = bx + c$, and $y = bx + d$ has an area of 18. The parallelogram bounded by the lines $y = ax + c$, $y = ax - d$, $y = bx + c$, and $y = bx - d$ has an area of 72. Given that $a$, $b$, $c$, and $d$ are positive integers, find the smallest possible value of $a + b + c + d$.

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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 $D$ be a point on side $[AB]$ of triangle $ABC$ with $|AB| = |AC|$, such that $[CD]$ is an angle bisector and $m(\widehat{ABC}) = 40^\circ$. Let $F$ be a point on the extension of $[AB]$ beyond $B$ such that $|BC| = |AF|$. Let $E$ be the midpoint of $[CF]$. If $G$ is the intersection of lines $ED$ and $AC$, what is the value of $x$ if $m(\widehat{FBG}) = x^\circ$?

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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 number of prime numbers $p$ such that $2p^4 - 7p^2 + 1$ is equal to the square of 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.

A list of integers has mode $32$ and mean $22$. The smallest number in the list is $10$. The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $m + 10$, the mean and median of the new list would be $24$ and $m + 10$, respectively. If $m$ were instead replaced by $m - 8$, the median of the new list would be $m - 4$. What is the value of $m$?

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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.

Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. Find the ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$. 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.

Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. Find the probability that the four chords are the sides of a convex quadrilateral. 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.

Three mutually tangent spheres of radius \(1\) rest on a horizontal plane. A sphere of radius \(2\) rests on them. What is the distance from the plane to the top of the larger sphere? Express your answer in the form \(3 + \frac{\sqrt{n}}{m}\) where \(n\) and \(m\) are integers. Please give the value of \(n + m\).

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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.

Find the integer that divides $3^{3n+1} + 5^{3n+2} + 7^{3n+3}$ for every positive integer $n$.

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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.

Find the radius of the inscribed circle of a triangle with sides of length \(13\), \(30\), and \(37\). Provide your answer in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) 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 geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. What is the common ratio? Provide your answer in the form of \(\frac{\sqrt{k} - m}{n}\) and 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.

Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000? 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.

A positive number is mistakenly divided by $6$ instead of being multiplied by $6$. Calculate the error committed, and provide the answer to the nearest percent.

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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 sum of all prime numbers between $1$ and $100$ that are simultaneously $1$ greater than a multiple of $4$ and $1$ less than a multiple of $5$. What is the sum?

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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 function $f$ satisfies the functional equation \[ f(x) + f(y) = f(x + y) - xy - 1 \] for every pair $x, y$ of real numbers. If $f(1) = 1$, find the number of integers $n \neq 1$ for which $f(n) = n$.

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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.

Real numbers \(x\) and \(y\) are chosen independently and uniformly at random from the interval \((0,1)\). Find the probability that \(\lfloor \log_2{x} \rfloor = \lfloor \log_2{y} \rfloor\), where \(\lfloor r \rfloor\) denotes the greatest integer less than or equal to the real number \(r\). 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.

Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. Find the difference in their money in dimes. The answer should be in the format $k(q - m)$. Please 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.

On a map with a scale of \( 400 \) miles to an inch and a half, a certain estate is represented by a rhombus with a \( 60^{\circ} \) angle. The diagonal opposite the \( 60^{\circ} \) angle is \( \frac{3}{16} \) inches. Calculate the area of the estate in square miles. 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.

在集合$\{0, 1, 2, \ldots, 2014\}$中，有多少个整数满足条件：$C^x_{2014} \ge C^{999}_{2014}$？请计算满足该条件的整数个数。

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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.

Label one disk "1", two disks "2", three disks "3", ..., fifty disks "50". Put these $1+2+3+ \cdots+50=1275$ labeled disks in a box. Disks are then drawn from the box at random without replacement. What is the minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same label?

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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.

$ABC$ is a triangle with vertices $A=(0,0)$, $B=(36,15)$, and $C=(x,y)$ where both $x$ and $y$ are integers. What is the minimum area of $\triangle ABC$? Provide your answer in the form of \(\frac{k}{m}\) where \(\frac{k}{m}\) 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.

What is the largest integer $n$ such that $2^n + 65$ is equal to the square of an integer? Find the value of $n$.

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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.

In $\triangle ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. $M$ is the midpoint of side $AB$, and $H$ is the foot of the altitude from $A$ to $BC$. If the length of $HM$ 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.

A cyclist travels at a constant speed of $22.0\,\text{km/hr}$ except for a $20$ minute stop. The cyclist’s average speed was $17.5\,\text{km/hr}$. How far did the cyclist travel? If the answer is $x\,\text{km}$, what is the value of $x \times 10$?

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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 smallest positive odd integer $n$ such that the product
\[2^{1/7} \cdot 2^{3/7} \cdot \ldots \cdot 2^{(2n+1)/7}\]
is greater than $1000$?
(In the product, the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.)

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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 non-congruent scalene triangles with perimeter $21$ have integer side lengths that form an arithmetic sequence? (In an arithmetic sequence, successive terms differ by the same amount.) 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.

How many positive integers are there such that $\left \lfloor \frac{m}{11} \right \rfloor = \left \lfloor \frac{m}{10} \right \rfloor$? ($\left \lfloor x \right \rfloor$ denotes the greatest integer not exceeding $x$.)

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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.

For how many pairs \((a,b)\) from \((1,2)\), \((3,5)\), \((5,7)\), \((7,11)\), does the polynomial \(P(x)=x^5+ax^4+bx^3+bx^2+ax+1\) have exactly one real root?

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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.

Horses X, Y, and Z are entered in a three-horse race in which ties are not possible. If the odds against X winning are $3\text{-to-}1$ and the odds against Y winning are $2\text{-to-}3$, what are the odds against Z winning? Express your answer in the form $k\text{-to-}m$. 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.

For how many positive integers $m$ is \[ \frac{2002}{m^2-2} \] a positive integer? Write your answer as a number.

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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 how many primes \( p \), there exist unique integers \( r \) and \( s \) such that for every integer \( x \), the congruence \( x^{3} - x + 2 \equiv (x - r)^{2} (x - s) \pmod{p} \) holds? Find the number of such 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.

If \(n\) is a multiple of \(4\), find the sum \(s=1+2i+3i^2+ \ldots +(n+1)i^{n}\), where \(i=\sqrt{-1}\). The original answer is in the form \(\frac{k}{m}(n+a-b\cdot i)\). Please calculate the value of \(k + m + a + b\).

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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.

In $\triangle PAT$, $\angle P=36^{\circ}$, $\angle A=56^{\circ}$, and $PA=10$. Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA}$, respectively, so that $PU=AG=1$. Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG}$, respectively. Find the degree measure of the acute angle formed by lines $MN$ and $PA$.

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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.

Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? Express your answer as a fraction in simplest form, in the format \(\frac{k}{m}\). 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 \(x, y \in (-2, 2)\) and \(xy = -1\), find the minimum value of \(u = \frac{4}{4-x^2} + \frac{9}{9-y^2}\). 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.

What are the last two digits of the decimal representation of \(9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}\)?

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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 the equation system \[\begin{array}{rcl} f(x) + g(x) &=& 0 \\ f(x)-(g(x))^3 &=& 0 \end{array}\] has more than one real root, where $a,b,c,d$ are reals and $f(x)=x^2 + ax+b$, $g(x)=x^2 + cx + d$, find at most how many distinct real roots the equation $f(x)g(x) = 0$ can have.

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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 \(a+b=b-c=c-a=3\), find the value of \(a+b+c\) 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.

The polynomial \(P(x) = x^3 + ax^2 + bx + c\) has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the \(y\)-intercept of the graph of \(y = P(x)\) is 2, what is the value of \(b\)?

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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.

The base of an isosceles triangle is $\sqrt{2}$. The medians to the legs intersect each other at right angles. If the area of the triangle is $x$, what is 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.

In their base $10$ representation, the integer $a$ consists of a sequence of $1985$ eights and the integer $b$ consists of a sequence of $1985$ fives. Find the sum of the digits of the base 10 representation of $9ab$.

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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, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. The original answer for the perimeter of the hexagon is in the form $k\sqrt{3}$. 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.

An *iterative average* of the numbers \(1\), \(2\), \(3\), \(4\), and \(5\) is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. Find the difference between the largest and smallest possible values that can be obtained using this procedure. The answer is in the form \(\frac{k}{m}\). 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.

If the area of \(\triangle ABC\) is \(64\) square units and the geometric mean (mean proportional) between sides \(AB\) and \(AC\) is \(12\) inches, find the value of \(\sin A\). The answer should be in the format \(\frac{k}{m}\), 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.

The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. Find how many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten.

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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.

Doug can paint a room in $5$ hours. Dave can paint the same room in $7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $t$ be the total time, in hours, required for them to complete the job working together, including lunch. The equation satisfied by $t$ is of the form \(\left(\frac{k}{m}+\frac{1}{n}\right)(t-p)=q\). Please find the value of \(k + m + n + p + q\).

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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.

Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. If Keiko's speed is \(x\) meters per second, the original answer is in the form \(\frac{k\pi}{m}\). 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.

The distance light travels in one year is approximately \( 5,870,000,000,000 \) miles. If the distance light travels in \( 100 \) years is expressed as \( x \times 10^{12} \text{ miles} \), what is the value 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.

A park is in the shape of a regular hexagon with sides of \(2\) km each. Starting at a corner, Alice walks along the perimeter of the park for a distance of \(5\) km. How many kilometers is she from her starting point? If the answer is \( \sqrt{x} \), what is the value 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.

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. If the answer is the total number of minutes from 1:00 PM to when she finishes the third task, what is the value of this number?

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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 $ABCD$ be a parallelogram with $\angle ABC=120^\circ$, $AB=16$, and $BC=10$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then find the length of $FD$.

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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 $y = \sqrt{x^2 - 2x + 1} + \sqrt{x^2 + 2x + 1}$, find the expression for $y$ in terms of $x$. The original answer is in the form |ax + b| + |cx + d|. Please give the value of a + b + c + d.

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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 $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated a third and a fourth time. Find the fractional part of the final mixture that is water. 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.

The sum of the distances from one vertex of a square with sides of length two to the midpoints of each of the sides of the square is in the form k + m\sqrt{n}. 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.

Vertex \(E\) of equilateral triangle \(ABE\) is in the interior of square \(ABCD\), and \(F\) is the point of intersection of diagonal \(BD\) and line segment \(AE\). If length \(AB\) is \(\sqrt{1+\sqrt{3}}\), then find the area of \(\triangle ABF\). The original answer is in the form \(\frac{\sqrt{k}}{m}\), 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 fraction \(\frac{\sqrt{a^2+x^2}-(x^2-a^2)/\sqrt{a^2+x^2}}{a^2+x^2}\) reduces to a simpler form. The reduced expression is in the form \(\frac{k a^2}{(a^2 + x^2)^{\frac{m}{n}}}\). Please 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.

A positive integer $n$ not exceeding $100$ is chosen such that if $n \leq 50$, the probability of choosing $n$ is $p$, and if $n > 50$, the probability of choosing $n$ is $3p$. What is the value of the probability that a perfect square is chosen multiplied by 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.

A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $\$17.71$. What was the cost of a pencil in cents?

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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 $a, b, c, d, e, f, g,$ and $h$ be distinct elements in the set:
\[
\{-7, -5, -3, -2, 2, 4, 6, 13\}.
\]
Find the minimum possible value of:
\[
(a + b + c + d)^2 + (e + f + g + h)^2
\]

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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 $ABC$ be a triangle and $AD$ be the bisector of the triangle ($D \in (BC)$). Assume that $AB = 14$ cm, $AC = 35$ cm, and $AD = 12$ cm. What is the area of triangle $ABC$ in cm$^2$? 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 $4 \times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?

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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 engineer said he could finish a highway section in $3$ days with his present supply of a certain type of machine. However, with $3$ more of these machines, the job could be done in $2$ days. If the machines all work at the same rate, how many days would it take to do the job with one machine?

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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 are exactly $N$ distinct rational numbers $k$ such that $|k| < 200$ and \[5x^2 + kx + 12 = 0\] has at least one integer solution for $x$. What is the value of $N$?

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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.

In how many ways can $7$ red and $7$ white balls be distributed into $7$ boxes such that every box contains exactly $2$ balls?

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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 $f$ be a function from the non-negative integers to the non-negative integers such that $f(nm) = n f(m) + m f(n)$, with the following conditions: $f(10) = 19$, $f(12) = 52$, and $f(15) = 26$. What is the value of $f(8)$?

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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.

Lines $L_1, L_2, \dots, L_{100}$ are distinct. All lines $L_{4n}$, where $n$ is a positive integer, are parallel to each other. All lines $L_{4n-3}$, where $n$ is a positive integer, pass through a given point $A$. Find the maximum number of points of intersection of pairs of lines from the complete set $\{L_1, L_2, \dots, L_{100}\}$.

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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.

What is the value of 100 times the expression $\sqrt{65} - \sqrt{63}$ rounded to the nearest 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.

How many pairs of real numbers $(x, y)$ are there such that \( x^{4} - 2^{-y^{2}} x^{2} - \lfloor x^{2} \rfloor + 1 = 0 \), where \( \lfloor a \rfloor \) denotes the greatest integer not exceeding \( a \)?

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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 polynomial functions $f$ of degree $\ge 1$ satisfy the following conditions?
\[
f(x^2) = [f(x)]^2 = f(f(x))
\]
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.

For how many ordered pairs of positive integers \((m,n)\), does \(m \cdot n\) divide \(2008 \cdot 2009 \cdot 2010\)? Express your answer as a product of powers of prime numbers in the form \(2^k \cdot 3^m \cdot 5^n\). 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.

In the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_1 = 400 \, \text{g}$, $M_2 = 200 \, \text{g}$, and $M_4 = 500 \, \text{g}$. What is the value of $M_3$ for the system to be in static equilibrium? If the answer is $x \, \text{g}$, what is the value 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.

A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. Find the probability that the distance from the point to the origin is at most two units. Express your answer in the form \frac{k}{m} where k and m are coprime, and 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.

Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. 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 rock is dropped off a cliff of height \(h\). As it falls, a camera takes several photographs at random intervals. In each picture, the distance the rock has fallen is measured. Let the average (expected value) of all these distances be \(kh\). If the number of photographs taken is huge, find \(k\). The original answer is in the form \(\frac{m}{n}\), where m and n are integers. Please give the value of 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.

Find the number of all positive integers $n$ such that $n + s(n) = 2016$, where $s(n)$ is the sum of all digits of $n$.

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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.

Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on, continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the third red light and the 21st red light? If the answer is x feet, what is the value of 10x?

**Note:** 1 foot is equal to 12 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.

Let \(a\) and \(b\) be positive reals such that:
\[ a = 1 + \frac{a}{b} \]
\[ b = 3 + \frac{4 + a}{b - 2} \]
The original answer is in the form \(\sqrt[3]{k}\). 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.

For how many integers $n$, are there four distinct real numbers satisfying the equation $|x^2 - 4x - 7| = n$?

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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.

$n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two numbers following it in the clockwise direction. If the sum of all numbers is $278$, find how many different values $n$ can take.

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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 sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Which digit is **not** included among the eight digits?

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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.

Suppose \(S\) cans of soda can be purchased from a vending machine for \(Q\) quarters. Find the expression that describes the number of cans of soda that can be purchased for \(D\) dollars, where \(1\) dollar is worth \(4\) quarters. The original answer is in \(\frac{kDS}{mQ}\) format, 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.

Let $ABCD$ be a trapezoid with $AB$ and $CD$ parallel, $\angle D = 2 \angle B$, $AD = 5$, and $CD = 2$. Find the length of $AB$.

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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 non-negative integer triples $(m,n,k)$ are there such that $m^3 - n^3 = 9^k + 123$?

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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 sequences of positive real numbers of the form $x, 2000, y, \ldots,$ in which every term after the first is 1 less than the product of its two immediate neighbors. Find the number of different values of $x$ for which the term 2001 appears somewhere in the sequence.

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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 five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $\overline{AB},\overline{BC},\overline{CD},\overline{DE},$ and $\overline{EA}$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?

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