Solve the following math problem step 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 $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay 1 coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.Write down the value when $k=4$.

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

Version 1. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, $$\sqrt[N]{\frac{x^{2N}+1}{2}}\leqslant a_n(x-1)^2+x.$$ Please provide the value of N when $a_n = N/2$.\n\nVersion 2. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, $$\sqrt[N]{\frac{x^{2N}+1}{2}}\leqslant b_N(x-1)^2+x.$$ Please provide the sum of  $a_{4}$$b_{6}$.

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Solve the following math problem step 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 all positive integers $n\geqslant2$ for which there exist $n$ real numbers $a_{1}<\cdots<a_{n}$ and a real number $r>0$ such that the $\frac{1}{2}n(n-1)$ differences $a_{j}-a_{i}$ for $1\leqslant i<j\leqslant n$ are equal, in some order, to the numbers $r^{1},r^{2},\ldots,r^{\frac{1}{2}n(n-1)}$. Please provide the sum of all such $n$ values.

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Solve the following math problem step 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 $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. For positive integer $n=8$, ﬁnd the smallest number of non-empty piles that one can obtain by performing a ﬁnite sequence of moves of this form.

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Solve the following math problem step 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 all triples of positive integers $(a, b, p)$ with $p$ prime such that $a^p=b!+p$. Please provide the sum of all elements in these triples.

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

Determine the maximal length $L$ of a sequence $a_{1}, \ldots, a_{L}$ of positive integers satisfying both the following properties:
- every term in the sequence is less than or equal to $2^{2023}$, and
- there does not exist a consecutive subsequence $a_{i}, a_{i+1}, \ldots, a_{j}$ (where $1 \leqslant i \leqslant j \leqslant L$ ) with a choice of signs $s_{i}, s_{i+1}, \ldots, s_{j} \in\{1,-1\}$ for which
$$
s_{i} a_{i}+s_{i+1} a_{i+1}+\cdots+s_{j} a_{j}=0
$$
Please provide the value of $n+1$ if the answer can be expressed in the form $2^{n} - 1$.

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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.

Let $n$ be a positive integer. We arrange $1+2+\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\text {th }}$ row contains exactly $i$ circles. The following figure shows the case $n=6$. In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two circles directly below it. In terms of $n$, find the largest value of $k$ such that if one circle from every row is coloured red, we can always find a ninja-path in which at least $k$ of the circles are red. Please provide the value of $k$ when $n = 6$.

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Solve the following math problem step 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 an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$. The original answer is in the form 4\frac{a}{b}m￥, 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.

Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$. Please provide the sum of all such primes $p$.

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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 all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers. Please provide the sum of all $a + b$ values from the solutions.

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

Solve the system $$\left\{\begin{array}{l}x^2+y^2-2z^2=2a^2,\\x+y+2z=4(a^2+1),\\z^2-xy=a^2.\end{array}\right.$$ Please provide the sum of the elements for all solutions when $a=1$.

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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.

Factor the expression $(b-c)^3+(c-a)^3+(a-b)^3$. The original answer is a product of three differences. Please provide the sum of the coefficients in the expression.

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Solve the following math problem step 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 an infinite cone, the measure of its unfolding's angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the greatest integer $n$ such that $n\cdot \alpha < 180^{\circ }$. Please provide the integer value directly.

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Solve the following math problem step 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 nonzero and nonequal integers $a, b, c$ so that $x( x- a) ( x- b) ( x- c) + 1$ factors into the product of two polynomials with integer coefficients. Provide the sum of $a, b, c$ if they are three consecutive 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.

Solve in integers the equation $$x+y=x^2-xy+y^2.$$ Please provide the sum of all x and y values from the solutions.

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Solve the following math problem step 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 three-digit numbers such that any of its positive integer powers ends with the same three digits and in the same order. Please provide the sum of all such numbers.

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Solve the following math problem step 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 all positive integer solutions of the equation $x^y=y^x\quad(x\neq y).$ Please provide the sum of all elements in the solutions.

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Solve the following math problem step 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 three figures after the decimal point in the decimal expression of the number \(\frac{0.123456789101112\ldots495051}{0.515049\ldots121110987654321}\). Please provide the value of the first three digits after the decimal point as a single 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 the decimal expression of a positive number, a, all decimals beginning with the third after the decimal point, are deleted (i.e., we take an approximation of a with accuracy to 0.01 with deficiency). The number obtained is divided by a and the quotient is similarly approximated with the same accuracy by a number b. What numbers b can be thus obtained? Please provide the sum of all possible values 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.

There are given segments $AB, CD$ and a point O such that no three of the points $A, B, C, D, O$ are on one straight line. The endpoint of a segment is marked if the straight line passing through it and O does not intersect another segment. How many marked endpoints are there? Please provide the sum of all possible numbers of marked endpoints.

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Solve the following math problem step 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 all positive integers n such that $(n − 1)!$ is not divisible by $n^2$. When p is equal to 1,please provide the sum of all 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.

Denote by $d(N)$ the number of divisors of N (1 and N are also considered as divisors of N). Find all N such that $N/d(N) = p$ is a prime. Please provide the sum of all possible values 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.

Number 4 has the following property: when divided by $q^2$, for any q, the remainder is less than $\frac {q^2}{2}$. Find all numbers with the same property and provide the sum of these numbers.

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

Is there a quadrilateral $ABCD$ of area 1 such that for any point O inside it the area of at least one of the triangles $\triangle OAB, \triangle OBC, \triangle OCD,$ or $\triangle OAD$ is an irrational number? The original answer involves a trapezoid with bases m and $\sqrt[n]{p}$. Please provide the sum of m，n and p involved in the original example.

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

Is there a real number h such that $[h\cdot1969^n]$ is not divisible by $[h\cdot1969^{n-1}]$ for any positive integer n? The original answer is in \frac{m^3}{n} format, 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.

Each pan of a balance has k weights, numbered from 1 to k. The left pan is heavier. It turns out that if we interchange the places of any two weights with the same number, then either the right pan becomes heavier or the two pans reach an equilibrium. For what k this is possible? Please provide the sum of all possible values 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.

A closed non-selfintersecting broken line is drawn on an infinite chessboard along the sides of its squares. There are K black squares inside the broken line. What is the greatest area of the figure bounded by the broken line when $K=4$?

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Solve the following math problem step 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 a triangle. The radius of the inscribed circle is equal to $\frac 43$ ; the lengths of the triangle’s heights are integers whose sum is equal to 13. Find the lengths of the triangle’s sides.
If the sum of the squares of all sides is in the form of $\frac{m}{n}$, find 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.

The numbers $1, 2, . . . , 1983$ stand at the vertices of a regular 1983-gon. Any of the axes of symmetry of the 1983-gon divides the numbers which do not stand at the vertices through which the axis passes (if any) into two sets: on either side of the axis. Let us call an arrangement of numbers good with respect to a given axis of symmetry if every number of one set is greater than the number symmetrical to it. How many such good arrangements exist with respect to any axis of symmetry? Please provide the number of such arrangements.

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Solve the following math problem step 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 the equation $\begin{aligned}\frac{x-49}{50}+\frac{x-50}{49}=\frac{49}{x-50}+\frac{50}{x-49}\end{aligned}$, if the sum of all possible values of $x$ is $\frac{b}{a}$, 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.

Suppose the "complexity" of a number is the minimum possible length of a sequence (if any) that begins with 0 and ends with that number, with each subsequent term either equal to half of the previous term or equal to the sum of the previous term and the previous term, which is equal to 1. (The length of the empty sequence is assumed to be 0.) Find the number with the largest "complexity" among all numbers of the form $\frac m {2^{50}}$, where $m = 1, 3, 5, . . . , 2^{50} − 1.$. If the number $\frac{a}{b}+\frac{c}{b}\cdot\left(\frac{1}{2}\right)^{50}$ has the largest "complexity", equal to 101, 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.

Given that in a convex quadrilateral $ABCD$, $\angle BAC=30^\circ, \angle ACD=40^\circ,\angle ADB=50^\circ,\angle CBD=60^\circ$ and $\angle ABC+\angle ADC>180^\circ.$, if $\angle A=\angle C=a^\circ, \angle B=b^\circ, \angle D=c^\circ.$, 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.

Given n points on a plane, no three of them lie on a straight line. There is a straight line through each pair of points. When $n=5$, what is the minimum number of pairs of non-parallel lines among these lines?

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Solve the following math problem step 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 fly is flying inside a regular tetrahedron with side a. The shortest distance for the fly to fly over every face and return to the initial position is $\frac{a}{\sqrt{m}}$. Find 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.

已知 $\triangle A B C$ 及其边 $B C$ 上的一点 $D$ 满足： $A B=2 B D$ ， $A C=3 C D$ ，且以 $A 、 D$ 为焦点可以作一个椭圆 $\Gamma$ 同时经过 $B 、 C$ 两点，求 $\Gamma$ 的离心率的$\sqrt{21}$倍。

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

已知抛物线 $y^{2}=2 p x$ 及定点 $A(a, b), B(-a, 0),\left(a b \neq 0, b^{2} \neq 2 p a\right). M$ 是抛物线上的点，设直线 $A M, B M$ 与抛物线的另一交点分别为 $M_{1}, M_{2}$ .
求证：当 $M$ 点在抛物线上变动时（只要 $M_{1}, M_{2}$ 存在且 $M_{1} \neq M_{2}$ ）直线 $M_{1} M_{2}$ 恒过一个定点.那么当a=3时，求出这个定点的横坐标.

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

给定正整数 $n=3$. 求最大的实数 $\lambda$, 使得不等式 $a_{n}^{2} \geqslant \lambda\left(a_{1}+a_{2}+\cdots+a_{n-1}\right)+2 a_{n}$ 对任何满足 $a_{1}<a_{2}<\cdots<a_{n}$ 的正整数 $a_{1}, a_{2}, \cdots, a_{n}$ 均成立.

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

设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的最小值的$\sqrt{3}$是 $\qquad$.

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

函数 $y=x+\sqrt{x^{2}-3 x+2}$ 的最小值

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

设集合 $S$ 是 $\{0,1, \cdots, 98\}$ 的 $m(m \geqslant 3)$元子集，满足对任意的 $x, y \in S$ ，均存在 $z \in$ $S$, 使得 $x+y \equiv 2 z(\bmod 99)$. 求 $m$ 的所有可能值。请给出所有可能值的和。

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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.

已知 $f(x)$ 是定义在 $(0,+
fty)$ 上的减函数。若 $f\left(2 a^{2}+a+1\right)<f\left(3 a^{2}-4 a+1\right)$成立，请给出 $a$ 的取值范围端点之和的整数部分。

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

若对一切 $\theta \in \mathbf{R}$ ，复数 $z=(a+\cos \theta)+(2 a-\sin \theta) i$ 的模不超过 2 ，则实数 $a$ 的取值范围的端点值之和。

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

已知复数 $Z_{1} 、 Z_{2}$ 满足 $\left|Z_{1}\right|=2,\left|Z_{2}\right|=3$, 若它们所对应的向量的夹角为 $60^{\circ}$, 则 $7\sqrt{133}\left|\frac{Z_{1}+Z_{2}}{Z_{1}-Z_{2}}\right|=$ $\qquad$

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Solve the following math problem step 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
$f(x)=\lg (x+1)-\frac{1}{2} \log _{3} x$.
(1) 解方程: $f(x)=0$;

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Solve the following math problem step 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 $a_{1}, a_{2}, \cdots, a_{10}$ 和 $b_{1}, b_{2}, \cdots, b_{10}$ 均为 $1,2, \cdots, 10$ 的排列，即集合
$$
\begin{aligned}
& \left\{a_{1}, a_{2}, \cdots, a_{10}\right\} \\
& =\left\{b_{1}, b_{2}, \cdots, b_{10}\right\} \\
& =\{1,2, \cdots, 10\}.
\end{aligned}
$$
记 $c_{k}=a_{k}^{2}+b_{k}^{2}(k=1,2, \cdots, 10)$. 设 $c_{1}$, $c_{2}, \cdots, c_{10}$ 中的最大数为 $M$, 最小数为 $m$. 求
 $M$ 的最小可能值;

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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.

Let $n$ be a positive integer. We arrange $1+2+\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\text {th }}$ row contains exactly $i$ circles. The following figure shows the case $n=6$. In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two circles directly below it. In terms of $n$, find the largest value of $k$ such that if one circle from every row is coloured red, we can always find a ninja-path in which at least $k$ of the circles are red. Please provide the smallest value of $n$ for which $k = 11$.

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

如果存在 $1,2, \cdots, n$ 的一个排列 $a_{1}, a_{2}, \cdots$, $a_{n}$, 使得 $k+a_{k}(k=1,2, \cdots, n)$ 都是完全平方数，则称 $n$ 为'好数'. 在集合 $\{11,13,15,17,19\}$ 中，哪些是'好数'，哪些不是'好数'？请提供'好数'和'不是好数'的个数之和。

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

在平面直角坐标系中，设点集
$\{P_{1},P_{2},\cdots,P_{4n+1}\}$
$=\left\{(x,y)|x,y\text{ 为整数},|x|\leq n,|y|\leq n,xy=0\right\}$,
其中$n\in\mathbf{N}_+.$求
$$(P_{1}P_{2})^{2}+(P_{2}P_{3})^{2}+\cdots+(P_{n}P_{4n+1})^{2}+(P_{4n+1}P_{1})^{2}$$
的最小值。请提供当 $n=1$ 时的最小值。

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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.

给定正整数 $n(n \geqslant 3)$. 对于 $1,2, \cdots, n$的任意一个排列 $P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$, 若 $i<j<k$, 则称 $x_{j}$ 介于 $x_{i}$ 和 $x_{k}$ 之间（如在排列（ $1,3,2,4$ ）中，3 介于 1 和 4 之间， 4 不介于 1 和 2 之间）。设集合 $S=\left\{P_{1}, P_{2}, \cdots, P_{m}\right\}$的每个元素 $P_{i}$ 是 $1,2, \cdots, n$ 的排列. 已知 $\{1,2, \cdots, n\}$ 的任意三个不同数中都有一个数，它在每个 $P_{i}(1 \leqslant i \leqslant m)$ 中都不介于另外两个数之间. 求 $m$ 的最大值。当 $n = 11$ 时，请给出答案。

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

给定素数 $p \geqslant 5$. 若 $p =7$，求三个连续正整数的乘积模 $p$ 的不同余数的个数。

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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.

求最大的实数 $C$ ，使得对任意的正整数 $n$ 和满足 $0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}=1$ 的数列 $\left\{x_{k}\right\}$ ，均有$$\sum_{k=1}^{n} x_{k}^{2}\left(x_{k}-x_{k-1}\right)>C$$。原始答案是\frac{m}{n}的形式，请给出m+n的值。

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

求所有的正整数对 $(x, y)$ ，满足 $x^{y}=y^{x-y}$. 请给出所有满足条件的整数对的所有数字之和。

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

已知凸六边形 $P$ 的所有顶点在一个单位正方形的边上, 且所有内角相等. 求 $P$ 的最短边长度的最大可能值。原始答案为$a-\frac{\sqrt{m}}{n}$的最简格式，请给出$a+m+n$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 $n(n \geqslant 2)$ 为给定的整数。求最小的实数 $\lambda$ ，使得对于任意实数 $x_{1}, x_{2}, \cdots, x_{n} \in$ $[0,1]$ ，存在 $\varepsilon_{1}, \varepsilon_{2}, \cdots, \varepsilon_{n} \in\{0,1\}$ ，满足：对
于任意的 $1 \leqslant i \leqslant j \leqslant n$ ，均有
$$
\left|\sum_{k=i}^{j}\left(\varepsilon_{k}-x_{k}\right)\right| \leqslant \lambda.
$$
若$n=3$，答案写为$\frac{m}{n}$，其中$m,n$互质，求$m+n$.

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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.

给定整数 $n \geqslant 2$ ，设实数 $x_{1}, x_{2}, \cdots, x_{n}$ 满足:
(1) $\sum_{i=1}^{n} x_{i}=0$;
(2) $\left|x_{i}\right| \leqslant 1(i=1,2, \cdots, n)$.
求 $\min _{1 \leqslant i \leqslant n-1}\left|x_{i}-x_{i+1}\right|$ 的最大值.
若 $n = 4$，最大值的形式为 $\frac{a}{b}$，其中$a,b$互质，求$a+b$.

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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.

已知 $0<x 、 y<1$. 求\(\frac{x y(1-x-y)}{(x+y)(1-x)(1-y)}\)的最大值。原始答案为\(\frac{m}{n}\)的格式，请给出m + n的值。

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

求所有的整数 $k$, 使得存在正整数 $a$和 $b$, 满足
$$\frac{b+1}{a}+\frac{a+1}{b}=k.$$ 请给出所有满足条件的整数 $k$ 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.

实数 $a_{1}, a_{2}, \cdots, a_{n}(n \geqslant 3)$ 满足
$a_{1}+a_{2}+\cdots+a_{n}=0$,
且 $2 a_{k} \leqslant a_{k-1}+a_{k+1}(k=2,3, \cdots, n-1)$.
求最小的 $\lambda(n)$, 使得对所有的 $k \in\{1$, $2, \cdots, n\}$ ，都有
$\left|a_{k}\right| \leqslant \lambda(n) \cdot \max \left\{\left|a_{1}\right|,\left|a_{n}\right|\right\}$。若$n=5$，答案格式为\frac{p}{q}，其中$p,q$互质，求$p+q$。

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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.

求所有的实数 $k$, 使得不等式
$$
a^{3}+b^{3}+c^{3}+d^{3}+1 \geqslant k(a+b+c+d)
$$
对任意 $a 、 b 、 c 、 d \in[-1,+\infty)$ 都成立. 原始答案为 \frac{m}{n} 的形式，请给出 m + n 的值。

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

求所有的正整数 $a$ ，使得对任意正整数 $n \geqslant 5$ ，均有 $\left(2^{n}-n^{2}\right) \mid\left(a^{n}-n^{a}\right)$ 。请给出所有满足条件的正整数$a$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.

已知凸六边形 $P$ 的所有顶点在一个单位正方形的边上，且所有内角相等. 求 $P$ 的最短边长度的最大可能值。请给出m + n的值，其中原始答案格式为1-\frac{\sqrt{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.

设 $x 、 y 、 z \in(0,1)$ ，满足  $\sqrt{\frac{1-x}{y z}}+\sqrt{\frac{1-y}{z x}}+\sqrt{\frac{1-z}{x y}}=2$.  求 $x y z$ 的最大值. 原始答案为 \frac{m}{n} 的形式，请给出 m + n 的值.

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

设 $n$ 为给定的正整数. 求最大的正整数 $k$ , 使得存在三个由非负整数组成的 $k$ 元集  \begin{aligned}  & A=\left\{x_{1}, x_{2}, \cdots, x_{k}\right\} ,\\  & B=\left\{y_{1}, y_{2}, \cdots, y_{k}\right\}, \\  & C=\left\{z_{1}, z_{2}, \cdots, z_{k}\right\}  \end{aligned}  满足对任意的 $j(1 \leqslant j \leqslant i)$ ，都有  $x_{j}+y_{j}+z_{j}=n$. 请提供当 n = 33 时的最大正整数 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.

已知 $a 、 b 、 c$ 为两两互质的正整数，且满足
a^{2}\left|\left(b^{3}+c^{3}\right), b^{2}\right|\left(a^{3}+c^{3}\right), c^{2} \mid\left(a^{3}+b^{3}\right).
求 $a 、 b 、 c$ 的值。请给出所有满足条件的三元组的元素和的总和。

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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 $\alpha 、 \beta \in\left(0, \frac{\pi}{2}\right)$. 求
$$
A=\frac{\left(1-\sqrt{\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2}}\right)^{2}}{\cot \alpha+\cot \beta}
$$
的最大值。原始答案为m+n\sqrt{k}的形式，请给出m+n+k的值

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

已知复数 $z_{1} 、 z_{2}$ 满足 $\left|z_{1}\right|<1$, $\left|z_{2}\right|<1, z_{1}+z_{2}$ 为实数，且 $z_{1}+z_{2}+z_{1} z_{2}=0$ 。求 $z_{1} z_{2}$ 的取值范围。请给出区间端点的倒数的和。

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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.

已知集合 $A=\left\{x \mid \log _{a}(a x-1)>1\right\}$ 。若 $3 \in A$, 请给出 a 的取值范围中所有有限端点的和的6倍。

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

若存在实数 $r$, 使得圆周 $x^{2}+y^{2}=r^{2}$ 上恰好有 $n$ 个整点, 则 $n$ 可以等于多少？请给出所有可能的n值之和。

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

已知存在实数 $r$, 使得圆周 $x^{2}+y^{2}=r^{2}$ 上恰好有 $n$ 个整点, 则 $n$ 可以等于（）。请提供所有满足条件的n值之和。

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

已知等差数列 $\left\{a_{n}\right\}$ 满足: $a_{1}<0, a_{90} \geqslant 74, a_{200}<200$, 且该数列在区间 $\left(\frac{1}{2}, 8\right)$ 中的项比在区间 $\left[14, \frac{43}{2}\right]$ 中的项数少 ,则$a_{4$的值为多少？

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

已知等差数列 $\left\{a_{n}\right\} 、\left\{b_{n}\right\}$ 的前 $n$ 项和分别为 $S_{n} 、 T_{n}$, 且对于一切正整数 $n$, 均有 $\frac{a_{n}}{b_{n}}=\frac{2 n-1}{3 n+1}$. 请给出 $\frac{S_{6}}{T_{5}}$ 的原始答案形式为 \frac{m}{n}，请计算 m + n 的值。

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

已知数列 $\left\{a_{n}\right\}$ 满足: $a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{N}^{*}\right)$ ，其中 $[x]$ 表示不超过实数 $x$ 的最大整数. 设 $C$ 为实数, 且对任意的正整数 $n$, 都有 $\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C$, 则 $C$ 的最小值是。原始答案为 \frac{a}{b} 格式，请给出 a + b 的值。

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

设点 $A$ 的坐标为 $(0,3)$, 点 $B, C$ 为圆 $O: x^{2}+y^{2}=25$ 上的两动点, 满足 $\angle B A C=90^{\circ}$. 求 $\triangle A B C$ 面积的最大值。原始答案为 \frac{m + n \sqrt{k}}{p} 格式，请给出 m + n + k + p 的值。

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

已知在三棱锥 $D-A B C$ 中, $\angle A C B=\angle A B D=90^{\circ}, C A=C B, \angle B A D=30^{\circ}$.若点 $C$ 在平面 $A B D$ 上的射影恰好在 $A D$ 上, 则二面角 $C-A B-D$ 的平面角的正弦值大小为$\frac{\sqrt{m}}{n}$形式，请给出m + n的值。

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

设函数 $f(x)=x-\ln (a x+2 a+1)+2$. 若对于任意 $x \geqslant-2$, 均有 $f(x) \geqslant 0$ 恒成立，则实数 $a$ 的取值范围为 \qquad. 请给出区间两端点之和的值。

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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.

过抛物线 $y=x^{2}$ 上两点 $A\left(1, y_{1}\right), B\left(x_{2}, y_{2}\right)$ 分别作抛物线的切线 $l_{1}, l_{2}, l_{1}$ 与 $l_{2}$交于点 $C$. 若 $\overrightarrow{A C} \cdot \overrightarrow{B C}=0$, 则 $x_{2}=$ 的原始答案为 \frac{m}{n} 的格式，请给出 m + n 的值。

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

设 $P(0, a)$ 是 $y$ 轴上异于原点的任意一点，过点 $P$ 且平行于 $x$ 轴的直线与曲线 $y=\frac{1}{a} \ln x$ 交于点 $Q$, 曲线 $y=\frac{1}{a} \ln x$ 在点 $Q$ 处的切线交 $y$ 轴于点 $R$, 则 $\triangle P Q R$ 的面积的最小值为。请给出原始答案 $\frac{\mathrm{e}^{\frac{m}{n}}}{\sqrt{k}}$ 的形式中 $m + n+k$ 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.

给定正整数 $a, b(a \leqslant b)$. 数列 $\left\{f_{n}\right\}$ 满足:
$$
f_{1}=a, f_{2}=b, f_{n+2}=f_{n+1}+f_{n}(n=1,2, \cdots)
$$
若对任意的正整数 $n$, 都有 $\left(\sum_{k=1}^{n} f_{k}\right)^{2} \leqslant \lambda \cdot f_{n} f_{n+1}$, 求实数 $\lambda$ 的最小值。原始答案是 $2 + \sqrt{5}$，请给出数字部分的和。

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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.

$x \in \mathbf{R}$, 用 $[x]$ 表示不超过 $x$ 的最大整数, 并用 $\{x\}=x-[x]$ 表示小数部分, 已知: $a_{1}=\sqrt{2}$, $a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}}$, 求 $\sum\limits_{k=1}^{2024} a_{k}$. 原始答案为 $a(\sqrt{b}+c)$ 的形式，请给出a+b+c的值。

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

在 $\triangle A B C$ 中, 若 $D$ 在 $B C$ 上, $A D$ 平分 $\angle B A C, A B=A D=3, C D=2$, 求 $\triangle A B C$ 的周长的两倍。

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Solve the following math problem step 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 $\overrightarrow{O A}$ 与 $\overrightarrow{O B}$ 的夹角为 $\theta,|\overrightarrow{O A}|=2,|\overrightarrow{O B}|=1, \overrightarrow{O P}=t \overrightarrow{O A}, \overrightarrow{O Q}=$ $(1-t) \overrightarrow{O B}$, 函数 $f(t)=|\overrightarrow{P Q}|$ 在 $t=t_{0}$ 时取最小值. 若 $0<t_{0}<\frac{1}{5}$, 求 $\theta$ 的取值范围。答案的标准形式为$\left(\frac{\pi}{a}, \frac{b \pi}{c}\right)$，请给出区间端点a+b+c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.

对 $1,2, \cdots, 2 n$ 的任意一个排列 $a_{1}, a_{2}, \cdots, a_{n}, b_{1}, b_{2}, \cdots, b_{n}$, 定义 $f(a_{1},a_{2},\cdots,a_{n}, b_{1},b_{2},\cdots,b_{n})=\sum^{n-1}\left|a_{i}b_{i}-a_{i+1}b_{i+1}\right|.$当 $a_{1}, a_{2}, \cdots, a_{n}, b_{1}, b_{2}, \cdots, b_{n}$ 取遍 $1,2, \cdots, 2 n$ 的所有排列时, 求 $f\left(a_{1}, a_{2}, \cdots, a_{n}, b_{1}, b_{2}, \cdots, b_{n}\right)=\sum^{n-1}\left|a_{i} b_{i}-a_{i+1} b_{i+1}\right|$的最小值。请给出 n = 6 时的结果。

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

n=1时，实数 $x_{i} \neq 0, i=1,2, \ldots, n$. 并且有 $\sum_{i=1}^{n} x_{i}=0$. 求下式的最小值.
$$
\left.\sum_{i=1}^{n} x_{i}^{2}\right)\left(\sum_{i=1}^{n} \frac{1}{x_{i}^{2}}\right)
$$

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

给定整数 $n>1$. 已知正整数 $m$ 和集合 $\{1,2, \ldots, n\}$ 的不同子集 $A_{1}, A_{2}, \ldots, A_{m}$ 满足对任意 $1 \leq i<j \leq n, A_{1}, A_{2}, \ldots, A_{m}$ 中恰好有 $n-j+i$ 个集合同时含有 $i, j$.求 $\sum\limits_{k=1}^{m}\left|A_{k}\right|^{2}$ 的最小可能值。当n=3时，答案为 $\frac{m}{n}$，请给出m+n的值。

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

在锐角三角形 $A B C$ 中, $A B$ 上的高 $C E$ 与 $A C$ 上的高 $B D$ 相交于点 $H$, 以 $D E$ 为直径的圆分别交 $A B 、 A C$ 于 $F 、 G$ 两点, $F G$ 与 $A H$ 相交于点 $K$, 已知 $B C=25, B D=20, B E=7$, 求 $25A K$ 的长。

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

将关于 $x$ 的多项式 $f(x)=1-x+x^{2}-x^{3}+\cdots-x^{19}+x^{20}$ 表为关于 $y$ 的多项式 $g(y)=a_{0}+a_{1} y+a_{2} y^{2}+\cdots$ $+a_{19} y^{19}+a_{20}{ }^{20}$, 其中 $y=x-4$, 则 $6(a_{0}+a_{1}+\cdots+a_{20})=$ $\qquad$ ;

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

将编号为 $1,2, \cdots, 9$ 的九个小球随机放置在圆周的九个等分点上，每个等分点上各有一个小球。设圆周上所有相邻两球号码之差的绝对值之和为 $S$。求使 $S$ 达到最小值的放法的概率的分子和分母之和。

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

已知植圆 $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ 的左右焦点分别为 $F_{1}$ 与 $F_{2}$, 点 $P$ 在直线 $l: x-\sqrt{3} y+8+2 \sqrt{3}=0$ 上. 当 $\angle F_{1} P F_{2}$取最大值时，比 $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}$ 的值为 \sqrt{m}-n 的形式，求 m - n 的值。

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

求函数 $y=\sqrt{x+27}+\sqrt{13-x}+\sqrt{x}$ 的最大和最小值。最大值为\(m\)，最小值为 \(n\sqrt{p}+\sqrt{q} \)的形式，请提供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.

已知函数 $f(x)=a \sin x-\frac{1}{2} \cos 2 x+a-\frac{3}{a}+\frac{1}{2}, a \in \mathbb{R}$ 且 $a \neq 0$ 。 (1)若对任意 $x \in \mathbb{R}$, 都有 $f(x) \leqslant 0$, 求 $a$ 的取值范围; (2)若 $a \geqslant 2$, 且存在 $x \in \mathbb{R}$, 使得 $f(x) \leqslant 0$, 求 $a$ 的取值范围。 请给出两问所有区间端点之和。

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

已知正三棱雉 $P-A B C$ 底面边长为 1, 高为 $\sqrt{2}$, 则其内切球半径的原始答案为\( \frac{\sqrt{m}}{n} \)的格式，请给出m + n的值。

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

若 $a, b \in \mathbf{R}_{+}$, 则 满 足 $\sqrt{x^{2}-\sqrt{2} a x+a^{2}}+$ $\sqrt{x^{2}-\sqrt{2} b x+b^{2}} \leqslant \sqrt{a^{2}+b^{2}}$ 的实数 $x$ 的取值范围是 \qquad 。请给出a=b=\sqrt2时的取值

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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.

已知实数 $x_{0} \in[0,1)$. 数列 $\left\{x_{k}\right\}$ 满足: 若 $x_{n-1}<\frac{1}{2}$, 则 $x_{n}=2 x_{n-1}$, 若 $x_{n-1} \geq \frac{1}{2}$, 则 $x_{n}=2 x_{n-1}-1(n=1,2, \cdots)$. 现知 $x_{0}=x_{2021}$, 则可能的 $x_{0}$ 的个数为 $\qquad$.答案的形式为m^{n}+k的形式，请给出m+n+k的值

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

已知复数 $z$, 满足 $\frac{z}{2}$ 与 $\frac{2}{z}$ 的实部和虚部均属于 $[-1,1]$, 则 $z$ 在复平面上形成轨迹的面积为原始答案为$a - b\pi$的形式，请给出a + b的值。

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

已知凸四边形 $A B C D$ 满足: $A B=1, B C=2, C D=4, D A=3$, 则其内切圆半径取值范围为 $\qquad$. 该区间端点的和形如\frac{a\sqrt{b}+c\sqrt{d}}{f}，请给出a+b+c+d+f的值

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

已知正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 1, 以顶点 $A$ 为球心， $\frac{2 \sqrt{3}}{3}$ 为半径作一个球，则球面与正方体的表面相交所得到的曲线的长等于原始答案的形式为 $\frac{m \sqrt{k}}{n} \pi$，请给出 m + n +k的值。

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

已知正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 1, 以顶点 $A$ 为球心， $\frac{2 \sqrt{3}}{3}$ 为半径作一个球，则球面与正方体的表面相交所得到的曲线的长等于 $\frac{m \sqrt{k}}{n} \pi$，请给出 m + n + 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.

解不等式
$$
\log _{2}\left(x^{12}+3 x^{10}+5 x^{8}+3 x^{6}+1\right)<1+\log _{2}\left(x^{4}+1\right).
$$
求出x的取值范围并给出该区间的端点和。

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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.

一种密码锁的密码设置是在正 $n$ 边形 $A_{1} A_{2} \cdots A_{n}$ 的每个顶点处赋值 0 和 1 两个数中的一个，同时在每个顶点处涂染红、蓝两种颜色之一，使得任意相邻的两个顶点的数字或颜色中至少有一个相同。假设 $n = 2$，请给出该种密码锁共有多少种不同的密码设置的值减去 4 的结果。

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

函数 $f(x)=a^{2 x}+3 a^{x}-2(a>0, a \neq 1)$ 在区间 $x \in[-1,1]$ 上的最大值为 8, 则它在这个区间上的最小值是。原始答案为 $-\frac{m}{n}$ 的形式，请给出 m + n 的值。

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

已知函数 $y=\left(a \cos ^{2} x-3\right) \sin x$ 的最小值为 -3 ，则实数 $a$ 的取值范围是。请提供该区间的两个端点之和的整数部分。

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

如果 $\cos ^{5} \theta-\sin ^{5} \theta<7\left(\sin ^{3} \theta-\cos ^{3} \theta\right), \theta \in[0,2 \pi)$, 那么 $\theta$ 的取值范围是 \qquad. 请给出 m + n + k + j 的和，其中原始答案格式为$\left(\frac{m\pi}{n}, \frac{k \pi}{j}\right)$。

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

给定 $n$ 个不同实数，其所有全排列组成的集合为 $A_{n}$ ，对于 $\left(a_{1}, a_{2}, \cdots, a_{n}\right) \in A_{n}$ ，若恰有两个不同的整数 $i, j \in\{1,2, \cdots, n-1\}$ 使得 $a_{i}>a_{i+1}, a_{j}>a_{j+1}$ 成立，则称该排列为 "好排列"。求 $A_{n}$ 中 "好排列"的个数。当 $n = 5$ 时，请给出 "好排列" 的个数。

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

直线 $x-2 y-1=0$ 与抛物线 $y^{2}=4 x$ 交于 $A, B$ 两点, $C$ 为抛物线上的一点, $\angle A C B=90^{\circ}$ ，则点 $C$ 的坐标和为多少？请提供两个可能坐标点的x坐标与y坐标之和。

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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.

正四棱雉 $P-A B C D$ 中, 侧面是边长为 1 的正三角形, $M, N$ 分别是边 $A B, B C$ 的中点, 则异面直线 $M N$ 与 $P C$ 之间的距离是 $\qquad$。原始答案是\frac{\sqrt{m}}{n}的形式，请给出m + n的值。

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

设椭圆 $\Gamma$ 的两个焦点是 $F_{1}, F_{2}$, 过点 $F_{1}$ 的直线与 $\Gamma$ 交于点 $P, Q$. 若 $\left|P F_{2}\right|=\left|F_{1} F_{2}\right|$,且 $3\left|P F_{1}\right|=4\left|Q F_{1}\right|$, 则椭圆 $\Gamma$ 的短轴与长轴的比值为。原始答案为\(\frac{m \sqrt{n}}{k}\)格式，请给出m + n + k的值。

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

设集合 $\left\{\left.\frac{3}{a}+b \right\rvert\, 1 \leqslant a \leqslant b \leqslant 2\right\}$ 中的最大元素与最小元素分别为 $M, m$ ，则 $M-m$ 的值为。原始答案为a-b\sqrt{c}的形式，请给出a+ b + c的值。

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

在平面直角坐标系 $x O y$ 中， $F_{1}, F_{2}$ 分别是椭圆 $\frac{x^{2}}{2}+y^{2}=1$ 的左、右焦点，设不经过焦点 $F_{1}$ 的直线 $l$ 与椭圆交于两个不同的点 $A, B$ ，焦点 $F_{2}$ 到直线 $l$ 的距离为 $d$ 。如果直线 $A F_{1}, l, B F_{1}$ 的斜率依次成等差数列，求 $d$ 的取值范围。答案标准形式为$(\sqrt{m}, n)$，请给出m+n的和的值。

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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.

若实数 $a, b, c$ 满足 $2^{a}+4^{b}=2^{c}, 4^{a}+2^{b}=4^{c}$, 求 $c$ 的最小整数值。标准答案格式为$log _{a} b-\frac{c}{d}$，请提供计算结果 a+b+c+d的值。

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

求具有下列性质的所有正整数 $k$ ：对任意正整数 $n, 2^{(k-1) n+1}$ 不整除 $\frac{(k n)!}{n!}$ 。请给出满足条件的所有 $k$ 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.

若实数 $\alpha, \beta, \gamma$ 构成以 2 为公比的等比数列， $\sin \alpha, \sin \beta, \sin \gamma$ 构成等比数列，则 $\cos \alpha$ 的值是以\frac{m}{n}格式给出，请给出m + n的值。

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

设正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 $1, \alpha$ 为过直线 $B D_{1}$ 的平面，则 $\alpha$ 截该正方体的截面面积的取值范围是 $\qquad$。答案的标准格式为$\left[\frac{\sqrt{a}}{b}, \sqrt{c}\right]$，请给出截面面积范围的两个端点a+b+c和的值。

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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.

Today, Ivan the Confessor prefers continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying $f(x)+f(y) \geq|x-y|$ for all pairs $x, y \in[0,1]$. Find the minimum of $\int_{0}^{1} f$ over all preferred functions. The original answer is in \frac{m}{n} format, 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.

已知点 $H$ 为 $\triangle A B C$ 的垂心，且满足 $\overrightarrow{H A}+2 \overrightarrow{H B}+6 \overrightarrow{H C}=0$ ，则角 $B=$
请给出答案的m + n的值，其中原始答案为\frac{m\pi}{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.

设正三棱锥 $P-A B C$ 的底面边长为 1 、高为 $\sqrt{2}$ ，过底边 $B C$ 作此三棱锥的截面.则截面面积的最小值为 $\frac{m \sqrt{n}}{k}$ 的形式，请给出$m + n + k$ 的值。

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

在半径为 1 的实心球中挖掉一个体积最大的圆锥，再将该圆锥重新融成一个圆柱，则圆柱表面积的最小值为 。原始答案的形式为 $\frac{m \sqrt[3]{n} \pi}{p}$，请给出 $m + n+p$ 的值。

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

已知 $x, y, z$ 为正实数, 且满足 $x^{\lg x} y^{\lg y} z^{\lg z}=5, x^{\lg y z} y^{\lg z x} z^{\lg x y}=2$, 则 $x y z$可能取值的乘积是多少 。

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

已知 $a_{1}=1, a_{n+1}=\lambda a_{n}^{2}+2\left(n \in \mathbf{N}^{*}\right)$, 若数列 $\left\{a_{n}\right\}$ 有上界, 即存在常数 $M$, 使得 $a_{n} \leqslant M$ 对 $n \in \mathbf{N}^{*}$ 恒成立，则实数 $\lambda$ 的最大值为 $\frac{m}{n}$ 的形式，请给出 m + n 的值。

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

设函数 $f(x)=2^{x}-2^{-x}$ 的反函数为 $y=f^{-1}(x)$, 则不等式 $\left|f^{-1}(x-1)\right|<1$ 的解集为一个区间$(a,b)$，求$a-b$。

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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.

数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1, \frac{a_{n+1}-a_{n}}{a_{n}}=\frac{a_{n+2}-a_{n+1}}{a_{n+2}}\left(n \in N^{*}\right)$, 若 $a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{6} a_{7}=3$, 则 $a_{2024}=$. 请给出原始答案为frac{m}{n}格式时m + n的值。

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

已知复数 $z$ 使得 $z-\frac{4}{z}$ 为纯虚数, 请给出 $|z-1-i|$ 的最小值的整数近似值。

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

在 $\triangle A B C$ 中, 已知 $\overrightarrow{A B} \cdot \overrightarrow{A C}=2 \overrightarrow{B C} \cdot \overrightarrow{B A}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$, 则 $\triangle A B C$ 最大角的正弦值为。原始答案为$\frac{m\sqrt{n}}{k}$的格式，其中$m,n,k$均为正整数，请给出$m + n+k$的值。

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

在 $\triangle A B C$ 中, $A, B, C$ 的对边分别为 $a, b, c$. 已知 $2 \sin ^{2} \frac{A+B}{2}=1+\cos 2 C$.若 $c^{2}=2 b^{2}-2 a^{2}$, 求 $\cos 2 A-\cos 2 B$ 的值. 原始答案为 $\frac{m}{n}$, 请给出 $m + n$ 的值.

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

已知两点 $A(-2,0), B(2,0)$, 动点 $P$ 在 $y$ 轴上的射影是 $H$, 且 $\overrightarrow{P A} \cdot \overrightarrow{P B}=$ $2|\overrightarrow{P H}|^{2}$ 。 已知过点 $B$ 的直线交曲线 $C$ 于 $x$ 轴下方不同的两点 $M, N$. 设 $M N$ 的中点为 $R$, 过 $R$ 与点 $Q(0,-2)$ 作直线 $R Q$, 若直线 $R Q$ 斜率是一个整数，求这个整数值.

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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.

抛物线 $y=\frac{1}{2} x^{2}$ 与直线 $l: y=x+4$ 所围成区域中有一个矩形 $A B C D$, 且点 $A, B$ 在抛物线上, 点 $D$ 在直线 $l$ 上, 其中点 $B$ 在 $y$ 轴右侧, 且 $|A B|=2 t(t>0)$.
当边 $C D$ 在直线 $l$ 上时,求矩形 $A B C D$ 面积的最大值.

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

设 $G(V, E)$ 是连通的简单图. 称 $E$ 的子集 $H$ 为 $G$ 的"偶度子图"，如果对任何 $x \in V, H$ 中共有偶数条边以 $x$ 为顶点. 设 $|V|=v,|E|=e$ ，问： $G$ 有多少个"偶度子图"？（注： $\emptyset$ 也是一个"偶度子图".）给出  $v=2$时的值。

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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.

给定正整数 $n$, 求最大的正整数 $k$, 使得如下命题成立: 对每个 $1 \leq i \leq 2 n$,设 $A_{i}$ 是若干个相邻的整数构成的集合（即每个 $A_{i}$ 都形如 $\{a+1, a+2, \cdots, a+r\}$ ，其中 $a$ 是整数, $r$ 是正整数），如果对任何 $1 \leq i \leq n, n+1 \leq j \leq 2 n$ 都有 $A_{i} \cap A_{j} \neq \emptyset$ ，则存在整数 $x$ ，使得集合 $\left\{1 \leq i \leq 2 n \mid x \in A_{i}\right\}$ 包含至少 $k$ 个不同的元素. 请给出当n=5时的k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.

Let $n = 10$. There are $n$ points in the plane, no three of them collinear. Each day, Tom erases one of the points, until there are three points left. On the $i$-th day, for $1 \leqslant i \leqslant n-3$, before erasing that day's point, Tom writes down the positive integer $v(i)$ such that the convex hull of the points at that moment has $v(i)$ vertices. Finally, he writes down $v(n-2)=3$. Find the greatest possible value that the expression  $$  |v(1)-v(2)|+|v(2)-v(3)|+\ldots+|v(n-3)-v(n-2)|  $$  can obtain among all possible initial configurations of $n$ points and all possible Tom's moves, when n is 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.

已知 \triangle A B C 为椭圆 \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 的内接三角形，且 A B 过点 P(1,0) ，则 \triangle A B C 的面积的最大值为 \frac{m\sqrt{q}}{n} 的形式，请给出 m + n + 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.

求 $\lim\limits _{n \rightarrow+\infty} \sum_{k=0}^{n-1} \frac{120}{\sqrt{n^{2}+k n}}$. 原始答案为 $m(\sqrt{n}-q)$ 的形式，请给出m+n+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.

若函数 $f(x)=x^{3}-3 x^{2}-t x-1, x \in R$ 有重根, 求 $t$ 的所有可能的值。答案为一个整数，基于原始答案$t=-3$ 或者 $\frac{m}{n}$，请给出-3 + 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.

二阶实矩阵 $X$ 满足 $X^{3}=I_{2}$, 请提供 $	ext{tr}(X)$ 的所有可能整数解之和。

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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.

设正四棱柱 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的底面 $A B C D$ 是单位正方形, 如果二面角 $A_{1}-B D-C_{1}$ 的大小是 $\frac{\pi}{3}$, 则 $A A_{1}=$ 的原始答案为 $\frac{\sqrt{m}}{n}$ 格式，请给出 $m + n$ 的值。

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

在 $\triangle A B C$ 中, 已知 $\overrightarrow{A B} \cdot \overrightarrow{A C}+2 \overrightarrow{B A} \cdot \overrightarrow{B C}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$。求 $\sin C$ 的最大值。原始答案为 $\frac{\sqrt{m}}{n}$ 的形式，请给出 m + n 的值。

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

已知正四棱雉 $V-A B C D$ 的高等于 $A B$ 长度的一半， $M$ 是侧棱 $V B$ 的中点， $N$ 是侧棱 $V D$ 上点，满足 $D N=2 V N$ ，则异面直线 $A M, B N$ 所成角的余弦值为\(\frac{\sqrt{m}}{n}\)的格式，请给出m + n的值。

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

设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的取值范围是一个区间，区间的两端点和形如\frac{a\sqrt{b}+c\sqrt{d}}{e}，请给出a+b+c+d+e

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

在 $\triangle A B C$ 中, 已知 $\overrightarrow{A B} \cdot \overrightarrow{A C}+2 \overrightarrow{B A} \cdot \overrightarrow{B C}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$。求 $\sin C$ 的最大值。原始答案的形式为$\frac{\sqrt{m}}{n}$，请给出m + n的值。

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

双曲线 $C$ 的方程为 $x^{2}-\frac{y^{2}}{3}=1$ ，左、右焦点分别为 $F_{1} 、 F_{2}$ 。过点 $F_{2}$ 作一直线与双曲线 $C$ 的右半支交于点 $P, Q$, 使得 $\angle F_{1} P Q=90^{\circ}$, 则 $\triangle F_{1} P Q$ 的内切圆半径的原始答案为 \sqrt{m} + n的形式，请给出m+n的值。

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

在平面直角坐标系 $x O y$ 中, 圆 $C_{1}: x^{2}+y^{2}-a=0$ 关于直线 $l$ 对称的圆为 $C_{2}: x^{2}+y^{2}+2 x-2 a y+3=0$, 请给出直线 $l$ 的方程系数之和。

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

已知 $\left\{a_{n}\right\}$ 是各项均为正数的等比数列，且 $a_{50}, a_{51}$ 是方程 $100 \lg ^{2} x=\lg (100 x)$的两个不同的解。求 $a_{1} a_{2} \cdots a_{100}$ 的值。已知原始答案为$\sqrt{m}$的形式，请给出m的值。

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

在平面直角坐标系 $x O y$ 中, 椭圆 $C$ 的方程为 $\frac{x^{2}}{9}+\frac{y^{2}}{10}=1, F$ 为 $C$ 的上焦点, $A$为 $C$ 的右顶点, $P$ 是 $C$ 上位于第一象限内的动点, 则四边形 $O A P F$ 的面积的最大值为 \qquad。原始答案为\frac{\sqrt{m} + n}{k}的格式，请给出m + n + k的值。

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

在平面直角坐标系 $x O y$ 中, 椭圆 $C$ 的方程为 $\frac{x^{2}}{9}+\frac{y^{2}}{10}=1, F$ 为 $C$ 的上焦点, $A$为 $C$ 的右顶点, $P$ 是 $C$ 上位于第一象限内的动点, 则四边形 $O A P F$ 的面积的最大值为。原始答案为\frac{m\sqrt{n}}{k}的格式，请给出m + n + k的值。

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

设 $f(x)$ 是定义在 $\mathbf{R}$ 上的函数, 若 $f(x)+x^{2}$ 是奇函数, $f(x)+2^{x}$ 是偶函数,则 $f(1)$ 的值为 \qquad. 原始答案为 $-\frac{m}{n}$ 的格式，请给出 $|m + n|$ 的值。

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

设复数 $z$ 满足 $|z|=1$, 使得关于 $x$ 的方程 $z x^{2}+2 \bar{z} x+2=0$ 有实根, 则这样的复数 $z$ 的和为 $-\frac{m}{n}$ 的形式，请给出 $m + n$ 的值。

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

在平面直角坐标系 $x O y$ 中, 设 $A B$ 是抛物线 $y^{2}=4 x$ 的过点 $F(1,0)$ 的弦, $\triangle A O B$ 的外接圆交抛物线于点 $P$ (不同于点 $O, A, B$ ). 若 $P F$ 平分 $\angle A P B$, 求 $|P F|$ 的所有可能值。原始答案为$\sqrt{m}-n$格式，请给出$m+n$的值。

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

已知圆锥的顶点为 $P$, 底面半径长为 $2$, 高为 $1$. 在圆锥底面上取一点 $Q$, 使得直线 $P Q$与底面所成角不大于 $45^{\circ}$ ，满足条件的点 $Q$ 所构成的区域的面积为 $m\pi$ 的形式，请给出 $m$ 的值。

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

在平面直角坐标系 $x O y$ 中, 圆 $\Omega$ 与抛物线 $\Gamma: y^{2}=4 x$ 恰有一个公共点, 且圆 $\Omega$ 与 $x$轴相切于 $\Gamma$ 的焦点 $F$ 。求圆 $\Gamma$ 的半径。原始答案为 $\frac{m \sqrt{3}}{n}$ 的格式，请给出 $m + n$ 的值。

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

平面直角坐标系中, $\vec{e}$ 是单位向量, 向量 $\vec{a}$ 满足 $\vec{a} \cdot \vec{e}=2$, 且 $|\vec{a}|^{2} \leqslant 5|\vec{a}+t \vec{e}|$对任意实数 $t$ 成立，求 $|\vec{a}|$ 的取值范围. 请给出该区间端点之和的整数部分。

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

设三棱雉 $P-A B C$ 满足 $P A=P B=3, A B=B C=C A=2$, 则该三棱雉的体积的最大值是多少。原始答案的格式为 $\frac{\sqrt{m} + n}{k}$ ，请给出 $m + n + k$ 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.

Let \(ABC\) be an equilateral triangle. \(D\) and \(E\) are midpoints of \([AB]\) and \([AC]\). The ray \([DE\) cuts the circumcircle of \(\triangle ABC\) at \(F\). The original answer is in the form \(\frac{\sqrt{k} - m}{n}\). 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.

Given \( 1 \le n \le 455 \) and \( n^3 \equiv 1 \pmod{455} \), find the number of solutions.

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Solve the following math problem step 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 distinct straight lines $OA$ and $OB$. From a point on $OA$, a perpendicular is drawn to $OB$; from the foot of this perpendicular, a line is drawn perpendicular to $OA$. From the foot of this second perpendicular, a line is drawn perpendicular to $OB$; and so on indefinitely. The lengths of the first and second perpendiculars are $a$ and $b$, respectively. Find the limit of the sum of the lengths of the perpendiculars as the number of perpendiculars grows beyond all bounds. The original answer is in the form \frac{k \cdot a^2}{k \cdot a - m \cdot 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.

The sum of the base-\( 10 \) logarithms of the divisors of \( 10^n \) is \( 792 \). 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.

Chloé chooses a real number uniformly at random from the interval \([0, 2017]\). Independently, Laurent chooses a real number uniformly at random from the interval \([0, 4034]\). Find the probability that Laurent's number is greater than Chloé's number. 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 is the sum of the last two digits of $7^{42} + 7^{43}$ in base $10$?

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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.

Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained by writing the integers from $1$ to $44$ in order, one after the other. Find the remainder when $N$ is divided by $45$.

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Solve the following math problem step 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 baseball is dropped on top of a basketball. The basketball hits the ground, rebounds with a speed of $4.0 \, \text{m/s}$, and collides with the baseball as it is moving downward at $4.0 \, \text{m/s}$. After the collision, the baseball moves upward as shown in the figure, and the basketball is instantaneously at rest right after the collision. The mass of the baseball is $0.2 \, \text{kg}$ and the mass of the basketball is $0.5 \, \text{kg}$. Ignore air resistance and any changes in velocities due to gravity during the very short collision times. If the speed of the baseball right after colliding with the upward-moving basketball is $x \, \text{m/s}$, 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.

How many distinct four-tuples $(a, b, c, d)$ of rational numbers are there such that:

$$a \log_{10} 2 + b \log_{10} 3 + c \log_{10} 5 + d \log_{10} 7 = 2005$$

Find the number of such distinct four-tuples.

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Solve the following math problem step 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 is inscribed in a triangle with side lengths $8$, $13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r < s$. What is the ratio $r:s$? Express your answer in the form of $x:y$ where $x$ and $y$ are integers. Please give the value of x + y.

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

In triangle $\triangle ABC$ with $|BC| > |BA|$, $D$ is a point inside the triangle such that $\angle ABD = \angle DBC$, $\angle BDC = 150^\circ$, and $\angle DAC = 60^\circ$. If the measure of $\angle BAD$ 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 number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$.

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Solve the following math problem step 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 equally spaced parallel lines intersect a circle, creating three chords of lengths $38$, $38$, and $34$. What is the distance between two adjacent parallel lines?

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

Phillip and Paula both pick a rational number, and they notice that Phillip's number is greater than Paula's number by $12$. They each square their numbers to get a new number, and see that the sum of these new numbers is half of $169$. Finally, they each square their new numbers and note that Phillip's latest number is now greater than Paula's by $5070$. What was the sum of their original numbers?

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Solve the following math problem step 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
\[ \log_3 7 \cdot \log_5 9 \cdot \log_7 11 \cdot \log_9 13 \cdots \log_{21} 25 \cdot \log_{23} 27 \]

Find the value of the expression.

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Solve the following math problem step 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 largest integer that divides each of the numbers in the sequence \(1^5 - 1, 2^5 - 2, 3^5 - 3, \cdots, n^5 - n, \cdots\).

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Solve the following math problem step 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 has vertices \(P_1, P_2, P_3, P_4, P_1', P_2', P_3'\), and \(P_4'\). Vertices \(P_2, P_3\), and \(P_4\) are adjacent to \(P_1\), and for \(1\leq i\leq 4\), vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'\), and \(P_1'P_4'\). Find the octahedron's side length. The original answer is in the form \(\frac{a\sqrt{b}}{c}\), 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.

Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. What digit is most likely to be the units digit of the **sum** of Jack's integer and Jill's 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 drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks, and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? 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 ladder is formed by removing some consecutive unit squares of a $10 \times 10$ chessboard such that for each $k$-th row ($k \in \{1,2,\dots, 10\}$), the leftmost $k-1$ unit squares are removed. Find the number of rectangles formed by the composition of unit squares that the ladder has.

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Solve the following math problem step 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 quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=92^{\circ}$ and $\measuredangle ADC=68^{\circ}$, find $\measuredangle EBC$. 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.

Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city?

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

Square \(ABCD\) has an area of \(36\), and \(\overline{AB}\) is parallel to the x-axis. Vertices \(A\), \(B\), and \(C\) are on the graphs of \(y = \log_{a}x\), \(y = 2\log_{a}x\), and \(y = 3\log_{a}x\), respectively. What is the value of \(a\)? The original answer is in the form \(k^{m/n}\), please give the value of k + m + n.

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

A three-quarter sector of a circle with a radius of \(4\) inches, together with its interior, can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? Provide your answer in the form of \(a\pi \sqrt{b}\), where \(a\) and \(b\) are integers. Please calculate 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.

Points $A, B, Q, D,$ and $C$ lie on the circle shown. The measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$ in degrees.

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Solve the following math problem step 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 $A$, $B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$. Find the radius of circle $B$. 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.

If $(0.2)^x = 2$ and $\log 2 = 0.3010$, then if the value of $x$ to the nearest tenth is $y$, what is the value of $10y$?

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Solve the following math problem step 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 $n$ be the number of ways that $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. 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.

How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?

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Solve the following math problem step 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 $AD$ be a median of $\triangle ABC$ such that $m(\widehat{ADB})=45^{\circ}$ and $m(\widehat{ACB})=30^{\circ}$. If the measure of $\widehat{ABC}$ 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.

Mary chose an even 4-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1, 2, \ldots, \frac{n}{2}, n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?

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Solve the following math problem step 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 table tennis tournament, every participant played every other participant exactly once. There were twice as many right-handed players as left-handed players. However, the number of games won by left-handed players was 40\% more than the number of games won by right-handed players. (Note: There were no ties and no ambidextrous players.) Find the total number of games played.

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

Triangles $ABC$ and $ABD$ are isosceles with $AB = AC = BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then what is the value of $\angle C + \angle D$ if the answer is in degrees?

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Solve the following math problem step 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 three representation of $x$ is
\[ 12112211122211112222. \]
What is the first digit (on the left) of the base nine representation 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.

For all positive numbers \(x, y, z\), evaluate the product:
\[
(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}]
\]
What is the simplified expression for this product? The simplified expression is in the form \(x^{-a}y^{-b}z^{-c}\). Please provide 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.

Given the function $f$ defined as follows: \[ f(0) = 0, \] \[ f(1) = 1, \] and for every integer $n \geq 1$, \[ f(3n-1) = f(n) - 1, \] \[ f(3n) = f(n), \] \[ f(3n+1) = f(n) + 1. \] Find the value of $f(2011)$.

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Solve the following math problem step 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 integer $x$, with $10 \le x \le 99$, is to be chosen. If all choices are equally likely, find the probability that at least one digit of $x$ is a $7$. The original answer is in the form \frac{k}{m}, where k and m are coprime. 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 $P(x) = x^{2022} + x^{1011} + 1$. Find the polynomial of the form ax^6 + bx^3 + c that divides $P(x)$. Please provide the sum of the coefficients 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.

In $\triangle ABC$, $\angle ABC = 45^\circ$. Point $D$ is on $\overline{BC}$ such that $2 \cdot BD = CD$ and $\angle DAB = 15^\circ$. Find the value of $\angle ACB$ in degrees. 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.

An unfair coin lands on heads with a probability of $\frac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. 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.

Points \(A(6,13)\) and \(B(12,11)\) lie on circle \(\omega\) in the plane. Suppose that the tangent lines to \(\omega\) at \(A\) and \(B\) intersect at a point on the \(x\)-axis. Find the value of \(\frac{\text{Area of }\omega}{\pi}\) in the format \(\frac{k}{m}\), and give the sum of k and m.

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

Find the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$. The answer is an integer. Please provide the integer value.

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Solve the following math problem step 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 such that angle $C$ is obtuse. Let $D \in [AB]$ and $[DC] \perp [BC]$. If $m(\widehat{ABC}) = \alpha$, $m(\widehat{BCA}) = 3\alpha$, and $|AC| - |AD| = 10$, find the length of $|BD|$.

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Solve the following math problem step 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 the infinite series \(1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} - \frac{1}{32} + \frac{1}{64} - \frac{1}{128} - \cdots\), let \(S\) be the (limiting) sum. The original answer is in \(\frac{k}{m}\) format, where \(k\) and \(m\) are integers. Please find the value of \(k + m\).

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

A semi-palindrome is a four-digit number whose first two digits and last two digits are identical. For instance, $2323$ and $5757$ are semi-palindromes, but $1001$ and $2324$ are not. Find the difference between the largest semi-palindrome and the smallest semi-palindrome.

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Solve the following math problem step 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 integer $n \geq 100$ such that $77$ divides $1+2+2^2+2^3+\dots + 2^n$? 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.

Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce a path length of $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.) If the answer is in the form of \(\arccos\frac{k}{m}\), 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.

Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{\left(\frac{n}{2}\right)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, find the sum of the 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.

A house and a store were each sold for $\$12000$. The house was sold at a loss of $20\%$ of its cost, and the store at a gain of $20\%$ of its cost. Find the entire transaction's net loss or gain in dollars.

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Solve the following math problem step 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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1) = 0$, $50 < f(7) < 60$, $70 < f(8) < 80$, and $5000k < f(100) < 5000(k+1)$ for some integer $k$. Find the integer $k$.

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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.

Points $A$ and $B$ lie on a circle centered at $O$, and $\angle AOB = 60^\circ$. A second circle is internally tangent to the first and tangent to both $\overline{OA}$ and $\overline{OB}$. The ratio of the area of the smaller circle to that of the larger circle is in the form \frac{k}{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.

In \(\triangle ABC\), we have \(AB = 1\) and \(AC = 2\). Side \(BC\) and the median from \(A\) to \(BC\) have the same length. The original answer is in the form \(\sqrt{k}\). Please determine 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.

Tom, Dick, and Harry started out on a $100$-mile journey. Tom and Harry traveled by automobile at a speed of $25$ mph, while Dick walked at a speed of $5$ mph. After a certain distance, Harry got off and continued walking at $5$ mph, while Tom went back for Dick and brought him to the destination at the same time that Harry arrived. Find the number of hours required for the trip.

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

If \(b > 1\), \(\sin x > 0\), \(\cos x > 0\), and \(\log_b \sin x = a\), find the value of \(\log_b \cos x\). The original answer is in the form \(\frac{m}{n} \log_b(1-b^{p})\). 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.

On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?

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

在\(\triangle ABC\)中，已知边长\(|AB|=3\)，且从点\(C\)作出的高的长度为\(2\)。求另外两条高的长度乘积的最大值。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。

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

How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5? 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.

The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$. If it costs $137.94$ to label all the lockers, how many lockers are there at the school? 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.

Consider two solid spherical balls:

1. **First Ball**:
   - Centered at $(0, 0, \frac{21}{2})$
   - Radius $6$

2. **Second Ball**:
   - Centered at $(0, 0, 1)$
   - Radius $\frac{9}{2}$

Determine the number of lattice points (points with integer coordinates $(x, y, z)$) that lie in the intersection of these two spheres.

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

We define a positive integer as "special" if it is divisible by all of its non-zero digits. Determine the maximum number of consecutive special numbers that can exist.

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Solve the following math problem step 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 largest integer not exceeding $\lfloor (n+1)a \rfloor - \lfloor na \rfloor$, where $n$ is a natural number and $a=\frac{\sqrt{2013}}{\sqrt{2014}}$.

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

A circular disk is divided by $2n$ equally spaced radii ($n > 0$) and one secant line. What is the maximum number of non-overlapping areas into which the disk can be divided? The original answer is in the form $kn + m$. 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.

A traffic light runs repeatedly through the following cycle: green for \(30\) seconds, then yellow for \(3\) seconds, and then red for \(30\) seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? Provide your answer in the form \(\frac{k}{m}\) 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.

The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{\text{th}}$ term of the sequence is $\log{b^n}$. 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.

The number of distinct pairs \((x,y)\) of real numbers satisfying both of the following equations:
\[ \begin{align*} x &= x^2 + y^2, \\ y &= 2xy \end{align*} \]
is \(\underline{\phantom{4}}\).

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

Juan and Mary play a two-person game where the winner gains 2 points and the loser loses 1 point. If Juan won exactly 3 games and Mary had a final score of 5 points, find the total number of games they played.

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

Regular octagon \(ABCDEFGH\) has area \(n\). Let \(m\) be the area of quadrilateral \(ACEG\). The answer is in the form \(\frac{\sqrt{k}}{l}\). Find the value of \(k + l\).

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Solve the following math problem step 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 $r$ be the speed in miles per hour at which a wheel, 11 feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{4}$ of a second, the speed $r$ is increased by 5 miles per hour. What is 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.

In Cartesian coordinates, two areas \(M\) and \(N\) are defined as follows:
- \(M: y \geq 0, y \leq x, y \leq 2-x\);
- \(N: t \leq x \leq t+1\).
Here, \(t\) is a real number such that \(t \in [0,1]\).

Determine the area of \(M \cap N\). The original answer is in the form \(-at^2 + bt + \frac{c}{d}\), where \(\frac{c}{d}\) is a simplified fraction. 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.

The graph of \( y = x^2 + 2x - 15 \) intersects the \( x \)-axis at points \( A \) and \( C \) and the \( y \)-axis at point \( B \). Find the value of \( \tan(\angle ABC) \). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).

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