sungyub/dapo-math-17k-verl huggingface dataset:
user: A player throwing a die scores as many points as on the top face of the die and is to play until his score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n$, and find the value of $\lim _{n\to\infty} p(n)$. The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 9
user: In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$ A A, A B, A C, B B, B C, A D, C C, B D, C D, D D $$ in which $A A$ means two seniors, $A B$ means a senior and a junior, etc. Determine numerical values to assign to $A, B, C, D$ so that the set of numbers $A + A, A + B, A + C, B + B$, etc., corresponding to the order above will be in descending magnitude. Please provide the sum of A + B + C + D for the least positive integers that satisfy this condition.

Answer: 23
user: For a fixed positive integer $n$ let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers satisfying $0 \leq x_{k} \leq 1$ for $k=1,2, \ldots, n$. Determine the maximum value, as a function of $n$, of the sum of the $n(n-1) / 2$ terms:
$$
\sum_{\substack{i, j=1 \\ i<j}}^{n}\left|x_{i}-x_{j}\right|.
$$
Please provide the value of this sum when n = 3.

Answer: 2
user: Determine all rational values for which $a, b, c$ are the roots of
$$x^{3}+a x^{2}+b x+c=0$$
How many pairs of \(a\), \(b\), \(c\) satisfy the given conditions?

Answer: 3
user: Find the volume of the four-dimensional hypersphere $x^{2}+y^{2}+z^{2}+t^{2}=r^{2}$, and also the hypervolume of its interior $x^{2}+y^{2}+z^{2}+t^{2}<r^{2}$. If \(r = 2\), given that the sum of these two parts can be written in the form of \(k\pi^{2}\), find the value of \(k\).

Answer: 24
user: 在平面直角坐标系中, 设点集 \( \{P_{1}, P_{2}, \cdots, P_{4 n+1}\} \) = \{(x, y) \mid x, y \text{ 为整数, }|x| \leqslant n,|y| \leqslant n, x y=0\}, 其中， \(n \in \mathbf{N}_{+}\)。求 \( \left(P_{1} P_{2}\right)^{2}+\left(P_{2} P_{3}\right)^{2}+\cdots+\left(P_{n} P_{4 n+1}\right)^{2}+\left(P_{4 n+1} P_{1}\right)^{2} \) 的最小值。请给出 n = 1 时的结果。

Answer: 8
user: Solve the system of differential equations$$\begin{aligned}& \frac{d x}{d t}=x+y-3 \\& \frac{d y}{d t}=-2 x+3 y+1\end{aligned}$$subject to the conditions $x=y=0$ for $t=0$.When \(t = \pi\), the value of \(y\) can be written in the form of \(a + be^{c\pi}\). Find the value of \(a + b + c\).

Answer: 2
user: Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_{1},\cdots, n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^4=B^7= ABA^{-1}B=1$, $A^2\neq1$, and $B\neq1$. How many elements of $G$ are of the form $C^2$ with $C$ in $G$?

Answer: 8
user: For each nonnegative integer $k$, let $d(k)$ denote the number of I's in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)-2$). Let $m$ be a positive integer. Express
$$\sum_{k=0}^{2^m-1}(-1)^{d(k)}k^m$$
in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials.Let \(m = 2\). Find the value of \(a+f(2)+g(2)\).

Answer: 5
user: Evaluate
$$\lim_{n \to \infty}\int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1}\cos^2 \{\frac{\pi}{2n}(x_{1}+x_{2}+\cdots x_{n})\}dx_{1}dx_{2}\cdots dx_{n}$$.
The original answer is in the format \(\frac{m}{n}\). Please provide the value of \(m + n\).

Answer: 3
user: Evaluate
$$\int_{0}^{\infty}\frac{\arctan(\pi x)-\arctan x}{x}dx$$.
The original answer is in the form of $(frac{m\pi}{n}})\ln\pi$. Please provide the value of $m + n$.

Answer: 3
user: Let $v$ be a vertex (corner) of a cube $C$ with edges of length 4. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$. The original answer is in the form $m - \frac{n\pi}{k}$, please provide the value of$ m + n+k$.

Answer: 15
user: Assume that the differential equation
$$
y^{\prime \prime \prime}+p(x) y^{\prime \prime}+q(x) y^{\prime}+r(x) y=0
$$
has solutions $y_{1}(x), y_{2}(x)$, and $y_{3}(x)$ on the whole real line such that
$$
y_{1}^{2}(x)+y_{2}^{2}(x)+y_{3}^{2}(x)=1
$$
for all real $x$. Let
$$
f(x)=\left(y_{1}^{\prime}(x)\right)^{2}+\left(y_{2}^{\prime}(x)\right)^{2}+\left(y_{3}^{\prime}(x)\right)^{2}
$$
Find constants $A$ and $B$ such that $f(x)$ is a solution to the differential equation
$$
y^{\prime}+A p(x) y=B r(x)
$$
The original answer is in the format $A = frac{m}{n}$,$B = frac{p}{q}$, please give the value of $m + n+p+q$.

Answer: 10
user: Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. The answer is in the form $r + s\sqrt{t}$ with $r$, $s$, and $t$ as positive integers. Please provide the value of $r + s + t$.

Answer: 27
user: A game of solitaire is played as follows. After each play, according to the outcome, the player receives either $a$ or $b$ points ($a$ and $b$ are positive integers with $a$ greater than $b$), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one of these is 58. Find the sum of $a$ and $b$.

Answer: 19
user: In how many ways can the integers from 1 to n be ordered subject to the condition that, except for the first integer on the left, every integer differs by 1 from some integer to the left of it? Please provide the number of ways for $n = 6$.

Answer: 32
user: Let 0<a<b. Evaluate $$\lim_{t \to 0} \left \{ {\int_{0}^{1} [bx+a(1-x)]^{t}dx}\right \} ^{1/t}$$[The final answer should not involve any operations other than addition, subtraction, multiplication, division, and exponentiation.] When \(a = 1\) and \(b = 2\), the limit can be written as \(\frac{a}{be}\), where both \(a\) and \(b\) are positive integers. Find the value of \(a + b\).

Answer: 5
user: Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$. and let $M_{n}$ denote the $2n \times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by $$\left.m_{ij}=\left\{\begin{array}{llll}x&\text{if}\quad i=j,\\a&\text{if}\quad i\neq j&\text{and}&i+j\text{is even},\\b&\text{if}\quad i\neq j&\text{and}&i+j\text{is odd}.\end{array}\right.\right.$$ Thus, for example, $M_2=\begin{pmatrix}x&b&a&b\\b&x&b&a\\a&b&x&b\\b&a&b&x\end{pmatrix}.$ Express $\lim_{x\rightarrow a}\det M_n/(x-a)^{2n-2}$ as a polynomial in $a,b$, and $n.$ where $\det M_n$ denotes the determinant of $M_n$. Please provide the value of this expression when $n = 3$, $a = 2$, and $b = 1$. The answer should be a single integer.

Answer: 27
user: Let $A(x, y)$ denote the number of points $(m, n)$ in the plane with integer coordinates $m$ and $n$ satisfying $m^{2}+n^{2} \leqslant x^{2}+y^{2}$. Let $g=\sum_{k=0}^{\infty} e^{-k^{2}}$. Express
$$
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(x, y) e^{-x^{2}-y^{2}} d x d y
$$
as a polynomial in $g$. The original answer is in the form of \(\pi(mg+n)^2\), please give the value of \(m+n\).

Answer: 1
user: Let $P$ be a non-self-intersecting closed polygon with $n$ sides. Let its vertices be $P_1$, $P_{2},\cdots,P_{n}.$ Iet $m$ other points$,Q_1,Q_{2},\cdots,Q_{m}$ interior to $P$ be given. Let the figure be triangulated. This means that certain pairs of the $(n+m)$ points $P_1,\cdots,Q_m$ are connected by line segments such that (i) the resulting figure consists exclusively of a set $T$ of triangles, (ii) if two different triangles in $T$ have more than a vertex in common then they have exactly a side in common, and (iii) the set of vertices of the triangles in $T$ is precisely the set of ($n+m)$ points $P_{1},\cdots,Q_{m}.$ How many triangles in $T?$ Please provide the number of triangles when n = 5 and m = 4.

Answer: 11
user: Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\ln a-b$, with $a$ and $b$ positive integers. Please provide the value of $a + b$.

Answer: 5
user: Express the sum \( \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n} \) as a rational number. The original answer is in the form of \( \frac{m}{n} \). Please provide the value of \( m + n \).

Answer: 11
user: Evaluate the following limit: \lim _{n \rightarrow \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos ^{2}\left\{\frac{\pi}{2 n}\left(x_{1}+x_{2}+\cdots x_{n}\right)\right\} d x_{1} d x_{2} \cdots d x_{n}. The original answer is in the format $\frac{m}{n}$, please give the value of $m + n$.

Answer: 3
user: Let R be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x + y + z\leqslant 1$. Let $w=1-x-y-z$. Express the value of the triple integral $$\iiint\limits_{R}x^{1}y^{9}z^{8}w^{4}dx dy dz$$ in the form $\frac{a!b!c!d!}{n!}$, where a, b, c, d, and n are positive integers. Please provide the integer result of this expression.Find the value of $a+b+c+d+n$.

Answer: 47
user: Let $A(x, y)$ denote the number of points ($m, n$) with integer coordinates $m$ and $n$ that satisfy $m^{2}+n^{2} \leqslant x^{2}+y^{2}$ in the plane. Let $g=\sum_{k=0}^{\infty} e^{-k^{2}}$. If
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(x, y) e^{-x^{2}-y^{2}} d x d y=\pi(ag-b)^{c}$$
Find the value of $a+b+c$.

Answer: 5
user: A game of solitaire is played as follows. After each play, according to the outcome, the player receives either a or b points (a and b are positive integers with a greater than b), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one of these is 58. Please provide the value of a + b.

Answer: 19
user: For which real numbers $a$, the sequence defined by the initial condition $u_{0}=a$ and the recursion $u_{n+1}=2u_{n}-n^{2}$ has $u_{n}>0$ for all $n>0$? Give the minimum value of $a$.

Answer: 3
user: Suppose A is a three-dimensional $a × b × c$ rectangular brick with $a, b, c > 0$. Suppose B is the set of all points whose distance from a point in A is at most 1 (in particular, B contains A), and the volume of B is $a b c+m(a b+a c+b c)+\pi(a+b+c)+\frac{n \pi}{p}$. Find the value of $m+n+p$.

Answer: 9
user: Let $V$ be the region in the Cartesian plane consisting of all points ( x , y ) satisfying the simultaneous conditions $$\left\vert x \right\vert\leqslant y\leqslant\left\vert x \right\vert+3 \quad and \quad y\leqslant4$$ Find the centroid ( $\overline{x}$ , $\overline{y}$ ) of $V$. The original answer for $\overline{y}$ is in $\frac{m}{n}$ format, please give the value of $m + n$.

Answer: 18
user: Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers. Please provide the value of a-b. Here $[x]$ is defined to be the integer such that $[x] \leqq x<[x]+1$ and $\log x$ is the logarithm of $x$ to base $e$.

Answer: 3
user: 给定一个 $2008 \times 2008$ 的棋盘，棋盘上每个小方格的颜色均不相同。在棋盘的每一个小方格中填入 $C 、 G 、 M 、 O$ 这 4 个字母中的一个, 若棋盘中每一个 $2 \times 2$ 的小棋盘中都有 $C 、 G 、 M 、 O$ 这 4 个字母, 则称这个棋盘为"和谐棋盘"。和谐棋盘有$a\times b^{c}-d$，求$a+b+c+d$的值。

Answer: 2046
user: 集合 $\left\{x \left\lvert\,-1 \leqslant \log _{-}^{1} 10<-\frac{1}{2}\right., x \in \mathbf{N}^{*}\right\}$ 的真子集的个数为$a^{b}-c$，求$a+b+c$的值。

Answer: 93
user: 设 $S_{n}=1+2+3+\cdots+n, n \in \boldsymbol{N}^{*}$, 求 $f(n)=\frac{S_n}{(n^2+32)S_n^{+}1}$的最大值。原始答案是 $\frac{m}{n}$ 的形式，请给出 $m + n$ 的值。

Answer: 51
user: $\sum\limits_{k=0}^{2022} C_{2022}^{k} \cos \frac{(1011-k) \pi}{2}$ 的值为$a^{b}$，求$a+b$的值。

Answer: 1013
user: 已知函数$f(x)=2\left(\sin ^{4} x+\cos ^{4} x\right)+m(\sin x+\cos x)^{4}$在 $x \in\left[0, \frac{\pi}{2}\right]$ 有最大值 5。请计算$m$的值加上5后的结果。

Answer: 6
user: For a positive integer $n$, let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$. Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$. Please provide the value of $n$.

Answer: 18
user: An unbiased coin is tossed $n$ times. $H$ is the number of heads and $T$ is the number of tails. What is the expected value of $|H-T|$? In other words, evaluate in closed form:
$$
\frac{1}{2^{n-1}} \sum_{k<n / 2}(n-2 k)\binom{n}{k}
$$
(In this problem, "closed form" means a form that does not involve a series. The given series can be reduced to a single term involving only binomial coefficients, rational functions of $n$ and $2^{n}$, and the largest integer function $[x]$.)
The expected value of $|H-T|$ is $\frac{b}{a}$ when $n=3$. Find the value of $a+b$.

Answer: 5
user: For each $t\geqslant0$, let $S_{t}$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$ f(1)-2 f(2 / 3)+f(1 / 3)\geqslant t[f(2 / 3)-2 f(1 / 3)+f(0)] . $$ Develop necessary and sufficient conditions on $t$ for $S_{t}$ to be closed under multiplication. Please provide the sum of the endpoints of the interval for which the condition holds.

Answer: 1
user: for all polygons $P$, where the double integral is taken over $H(P)$. The original answer is in $\frac{m}{n}$ format, please give the value of $m + n$.

Answer: 11
user: A convex pentagon $P = ABCDE$, with vertices labeled consecutively, is inscribed in a circle of radius 1. Find the maximum area of P subject to the condition that the chords AC and BD be perpendicular. The original answer is in the format $1 + \frac{m}{n} \sqrt{3}$, please give the value of $m + n$.

Answer: 7
user: Assume $F$ is a finite field with an odd number of elements $m$. Assume $p(x)$ is an irreducible (i.e., non-factorizable) polynomial in $F$ of the form
$$
x^{2}+b x+c, \quad b, c \in F
$$
For how many elements $k$ in $F$, $p(x)+k$ is irreducible in $F$? How many elements are there when $m=7$.

Answer: 3
user: Let $u, f$, and g be functions defined for all real numbers x, such that

$$
\frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x).
$$

If $u(x)=g(x+a)-f(x+b)+f(x+c)-f(x+d)+f(x+e)$, what is the value of $a+b+c+d+e$.

Answer: 20
user: Evaluate the infinite product
$$
\prod_{n=2}^{\infty} \frac{n^{3}-1}{n^{3}+1}
$$
The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 5
user: Determine all polynomials $P(x)$ such that $P\left(x^{2}+1\right)=(P(x))^{2}+1$ and $P(0)=0$. Please provide the minimum positive integer value $P (x) $that meets the criteria.

Answer: 1
user: Establish necessary and sufficient conditions on the constant $k$ for the existence of a continuous real valued function $f(x)$ satisfying $f(f(x))=k x^{9}$ for all real x. Provide the smallest integer value of k that satisfies the condition.

Answer: 0
user: For each $t\geqslant0$, let $S_t$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$f(1)-2f(2/3)+f(1/3)\geqslant[f(2/3)-2f(1/3)+f(0)]$$. Develop necessary and sufficient conditions on $t$ for $S_t$ to be closed under multiplication. Please provide the value of a + b if the original answer is in the format $a\leqslant t\leqslant b$.

Answer: 1
user: Let r and s be positive integers. Derive a formula for the number of ordered quadruples $(a, b, c, d)$ of positive integers such that
$$3^{r}·7^{s}=
\mathrm{lcm}[a,b,c]=\mathrm{lcm}[a,b,d]=\mathrm{lcm}[a,c,d]=\mathrm{lcm}[b,c,d]$$.
Please provide the value of the formula when r = 2 and s = 1. The answer should be the sum of the coefficients of the expanded expression of the original answer.

Answer: 1089
user: Find positive integers $n$ and $a_{1}, a_{2}, \ldots, a_{n}$ such that \n$$\na_{1}+a_{2}+\cdots+a_{n}=1979\n$$\nand the product $a_{1} a_{2} \cdots a_{n}$ is as large as possible. Please provide the value of $a + n$.

Answer: 662
user: Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0<x<1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that
$$
u \leqslant \int_{0}^{1}\left|f^{\prime}(x)-f(x)\right| d x$$
for all $f$ in $C$. The original answer is in \(\frac{m}{ne}\) format, please provide the value of m+ n.

Answer: 2
user: A point P moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let $N(T)$ be the number of starting directions from a fixed interior point $P_{0}$ for which P escapes within T units of time. Find the least constant a for which constants b and c exist such that $$ N(T) \leq a T^{2}+b T+c $$ for all $T>0$ and all initial points $P_{0}$. The original answer is in \(\pi\) format, please give the integer part of the answer.

Answer: 1
user: Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. The answer is in the form $r+s \sqrt{t}$ with $r, s$, and $t$ being positive integers. Please provide the value of $r + s + t$.

Answer: 27
user: Let $p$ be a prime number. Let $J$ be the set of all $2 \times 2$ matrices $\binom{a b}{c d}$ whose entries are chosen from $\{0,1,2, \cdots, p-1\}$ and satisfy the conditions $a+d \equiv 1(\bmod p), a d-b c \equiv 0(\bmod p)$. Determine how many members $J$ has. The original answer is in the format of $p^2 + p$. Let p=5，Determine how many members $J$ has.

Answer: 30
user: Evaluate $$\sum_{k=0}^{n} (-1)^k\binom{n}{k} (x-k)^n.$$ Provide the integer value of n=3.

Answer: 6
user: Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i](i=1, 2,3)$. If the distribution of each random number is uniform with respect to length in the interval it is chosen from, determine the expected value of the smallest of the three numbers chosen. The original answer is in the format $\frac{a}{b} L_{1}^{2} L_{2} L_{3}-\frac{c}{d} L_{1}^{3}\left(L_{2}+L_{3}\right)+\frac{e}{f} L_{1}^{4}$, please give the value of a+b+c+d+e+f.

Answer: 23
user: Let $F(x)$ be a real valued function defined for all real $x$ except for $x=0$ and $x=1$ and satisfying the functional equation $F(x)+F\{(x-1) / x\}=1+x$. Find all functions $F(x)$ satisfying these conditions. The original answer is in the format \frac{m}{n}, please provide the value of m + n when x = 2.

Answer: 7
user: A particle moving on a straight line starts from rest and attains a velocity $v_0$ after traversing a distance $s_0$. If the motion is such that the acceleration was never increasing, find the maximum time for the traverse. 假设So=2,Vo=1,The original answer is in the format \frac{m}{n}, please give the value of m+n.

Answer: 5
user: Let u, f, and g be functions, defined for all real numbers x, such that $$\frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x).$$ Determine u(x) in terms of f and g. Please provide the value of u(x) when x = 1. The answer should be in the format $u(x)=g(a)-f(b)+f(c)-f(d)+f(e)$, directly give the value of a+b+c+d+e.

Answer: 25
user: Determine the minimum value of
$$(r-1)^{2}+\left(\frac{s}{r}-1\right)^{2}+\left(\frac{t}{s}-1\right)^{2}+\left(\frac{4}{t}-1\right)^{2}$$
for all real numbers r, s, t with $1 \leq r \leq s \leq t \leq 4$.The original answer form is $a-b \ sqrt {c} $, find the value of a+b+c

Answer: 22
user: Let $ABC$ be a triangle with angle $A< \angle C<90^\circ < \angle B$. Consider the bisectors of the external angles at $A$ and $B$, each measured from the vertex to the opposite side (extended). Suppose both of these line-segments are equal to $AB$. Compute the sum of angles $A$ and $B$. 请给出角A和角B的和。

Answer: 24
user: (a) Find a solution that is not identically zero, of the homogeneous linear differential equation $$( 3x^2+x-1 ){y}''- ( 9x^2+9x-2 ){y}'+( 18x+3 )y=0$$ Intelligent guessing of the form of a solution may be helpful. 
(b) Let $y=f(x)$ be the solution of the nonhomogeneous differential equation $$( 3x^2+x-1){y}''-( 9x^2+9x-2){y}'+( 18x+3)=6( 6x+1) $$ that has $f(0)=1$ and $(f(-1)-2)(f(1)-6)=1$. Find integers $a, b, c$ such that $(f(-2)-a)(f(2)-b)=c$. Please provide the value of a + b + c.

Answer: 21
user: Assume that the differential equation $$y^{\prime\prime\prime}+p(x)y^{\prime\prime}+q(x)y^{\prime}+r(x)y = 0$$ has solutions $y_1(x)$, $y_2(x)$, and $y_3(x)$ on the whole real line such that $$y_{1}^{2}(x)+y_{2}^{2}(x)+y_{3}^{2}(x)=1$$ for all real x. Let $$f(x)=\left(y_{1}^{\prime}(x)\right)^{2}+\left(y_{2}^{\prime}(x)\right)^{2}+\left(y_{3}^{\prime}(x)\right)^{2}$$. Find constants $A$ and $B$ such that $f(x)$ is a solution to the differential equation $$y^{\prime}+Ap(x)y=Br(x)$$. The original answer is in the form of \frac{m}{n}. Please provide the value of m + n.

Answer: 5
user: A sequence of convex polygons $\{P_n\}$, $n\geqslant0$, is defined inductively as follows. $P_0$ is an equilateral triangle with sides of length 1. Once $P_n$ has been determined, its sides are trisected; the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$. Thus, $P_{n+1}$ is obtained by cutting corners off $P_n$, and $P_n$ has $3\cdot2^n$ sides. ($P_1$ is a regular hexagon with sides of length $1/3$.) Express $\lim_{n\rightarrow\infty}Area(P_n)$ in the form $\sqrt{a}/b$, where a and b are positive integers. Please provide the value of a + b.

Answer: 10
user: Let $V$ be the region in the cartesian plane consisting of all points $(x, y)$ satisfying the simultaneous conditions $$|x|\leqslant y\leqslant|x|+3\text{ and }y\leqslant4.$$ Find the centroid $(\bar{x},\bar{y})$ of $V$. The original answer for $\bar{y}$ is in the form \(\frac{m}{n}\). Please provide the value of m + n$.

Answer: 18
user: (a) Let $z$ be a solution of the quadratic equation
$$
a z^{2}+b z+c=0
$$
and let $n$ be a positive integer. Show that $z$ can be expressed as a rational function of $z^{n}, a, b, c$.
(b) Using (a) or by any other means, express $x$ as a rational function of $x^{3}$ and $x+(1 / x)$. The original answer is in \frac{m}{n} format, please give the value of m + n if x=v=2.

Answer: 13
user: Express
$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n}$$
as a rational number. The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.

Answer: 11
user: Evaluate $$\lim\limits_{x\to1^-}\prod\limits_{n=0}^\infty\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}.$$ Please provide the value of a-b, where the original answer is in the form of \ln a-b.

Answer: 1
user: Determine all real numbers $a>0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region $$R=\{(x,y);0\leq x\leq a,0\leq y\leq f(x)\}$$ has perimeter $k$ units and area $k$ square units for some real number $k.$ Please provide the smallest integer value of $a$ that satisfies the condition.

Answer: 2
user: Evaluate $\int_{0}^{1}\frac{ln(x+1)}{x^2+1}dx$. The original answer is in the format $\frac{m\pi}{n}$, please provide the value of $m + n$.

Answer: 10
user: Let $S={(a, b) \mid a=1,2,..., n, b=1,2,3}$. A rook tour of $S$ is a polygonal path made up of line segments connecting points $p_1, p_2,\dots,p_{3n}$ in sequence such that (i) $p_i \in S$, (ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1 \le i< 3n$, (iii) for each $p\in S$ there is a unique i such that $p_i =p$. How many rook tours are there that begin at $(1,1)$ and end at $(n,1)$? Please provide the value of the number of rook tours when n = 5.

Answer: 8
user: Let $a_1, a_2, \ldots, a_n$ be real numbers, and let $b_1, b_2, \ldots, b_n$ be distinct positive integers. Suppose there is a polynomial $f(x)$ satisfying the identity $$(1-x)^n f(x)=1+\sum_{i=1}^n a_i x^{b_i}.$$ Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \ldots, b_n$ and $n$ (but independent of $a_1, a_2, \ldots, a_n$). Provide the value of $b_1 + b_n + n$ for $b_1 = 2$, $b_n = 3$, and $n = 2$.

Answer: 7
user: Evaluate $\sum_{n=0}^{\infty} \operatorname{Arccot}\left(n^2+n+1\right)$, where $\operatorname{Arccot} t$ for $t \geqslant 0$ denotes the number $\theta$ in the interval $0<\theta \leqslant \pi / 2$ with $\cot \theta=t$. The original answer is in the format \frac{m\pi}{n}, please give the value of m + n.

Answer: 3