ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Determine the largest real number $a$ such that for all $n\geqslant1$ and for all real numbers $x_0,x_1,\ldots,x_n$ satisfying $0=x_0<x_1<x_2<\cdots<x_n$, we have $$\begin{aligned}\frac{1}{x_1-x_0}+\frac{1}{x_2-x_1}+\cdots+\frac{1}{x_n-x_{n-1}}\geqslant a\left(\frac{2}{x_1}+\frac{3}{x_2}+\cdots+\frac{n+1}{x_n}\right).\end{aligned}\quad(1)$$ The largest $a$ is in the format of $\frac{m}{n}$. Please provide the value of$ m + n$.
Answer: 13
Problem: The leader of an IMO team chooses positive integers $n$ and $k$ with $n>k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader's in exactly $k$ positions. (For example, if $n=3$ and $k=1$, and if the leader chooses $101$, the deputy leader would write down $001,111$and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader's string. What is the minimum number of guesses (in terms of $n$ and $k)$ needed to guarantee the correct answer? Please give the sum of guesses when $n = 6$ and $k = 3$.
Answer: 2
Problem: Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n).$ Please provide the value of $f(n)$ when $n=2$. Directly give the value of $f(n)$.
Answer: 4
Problem: Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: $\bullet$ In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. $\bullet$ In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the $frst\textit{line. }$ $\bullet$ In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a,b,c,d$ are four (not necessarily distinct) numbers from the $frst\textit{line. }$ Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line. Please provide the sum of all possible values of $q$.
Answer: 0
Problem: For each permutation $y_1,y_2,\cdots,y_n$ of these numbers we have $\sum\limits_{i=1}^{n-1}y_iy_{i+1}=y_1y_2+y_2y_3+y_3y_4+\cdots+y_{n-1}y_n\geqslant-1$. Find the largest constant $K=K(n)$ such that $\sum\limits_{1\leqslant i<j\leqslant n}x_ix_j\geqslant K$ holds for every Shiny$n$-tuple $(x_1,x_2,\cdots,x_n)$. Please provide the value of $K$ when $n=3$.
Answer: 1
Problem: Sir Alex plays the following game on a row of $9$ cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: (1) Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. (2) Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j.$ Replace the number in one of the cells with $2^j+1$ and erase the number in the other cell. At the end of the game, one cell contains the number $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. Please provide the value of the maximum number of moves when $n = 10$.
Answer: 2025
Problem: Let $n>1$ be an integer. An $n\times n\times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one color. For each $n\times n\times1$ box consisting of $n^2$ unit cubes (of any of the three possible orientations), we consider the set of the colors present in that box (each color is listed only once). This way, we get $3n$ sets of colors, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colors that are present. Please provide the maximum number of colors when n = 6.
Answer: 91
Problem: Let $n$ be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The neighborhood of a lattice point $c$ consists of all lattice points within the axis-aligned $(2n+1)\times (2n+1)$ square centered at $c$, apart from $c$ itself. We call a butterfly $lonely, crowded$, or $comfortable$, depending on whether the number of butterflies in its neighborhood $N$ is respectively less than, greater than, or equal to half of the number of lattice points in $N.$ Every minute, all lonely butterflies fly away simultaneously. This process goes on for as long as there are any lonely butterflies. Assuming that the process eventually stops, determine the number of comfortable butterflies at the final state. Please provide the number of comfortable butterflies when $n = 3$.
Answer: 10
Problem: Find all pairs $(p,q)$ of prime numbers with $p>q$ for which the number $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$ is an integer. Please provide the value of $p+q$ for such pairs.
Answer: 5
Problem: Let $a_0, a_1, a_2, \ldots$ be a sequence of real numbers such that $a_0=0, a_1=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying
$$
a_n=\frac{a_{n-1}+\cdots+a_{n-k}}{k}
$$
Find the maximal possible value of $a_{2018}-a_{2017}$. The original answer is in the form \frac{m}{n^2}, please give the value of $m + n$.
Answer: 4033
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 $k=4$.
Answer: 134
Problem: A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold: (i) each triangle from $T$ is inscribed in $\omega$; (ii) no two triangles from $T$ have a common interior point. Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$. Please provide the maximum value of $t$.
Answer: 4
Problem: Find all pairs $(m, n)$ of positive integers satisfying the equation $$ \begin{equation*} \left(2^{n}-1\right)\left(2^{n}-2\right)\left(2^{n}-4\right) \cdots\left(2^{n}-2^{n-1}\right)=m! \tag{1} \end{equation*} $$
Please provide the sum of all integers in the pairs $(m, n)$.
Answer: 7
Problem: Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. Please provide the sum of all integers in the solutions.
Answer: 18
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}$.
Answer: 5
Problem: Let $n$ be an integer with $n \geqslant 2$. On a slope of a mountain, $n^2$ checkpoints are marked, numbered from 1 to $n^2$ from the bottom to the top. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars numbered from 1 to $k$; each cable car provides a transfer from some checkpoint to a higher one. For each company, and for any $i$ and $j$ with $1 \leqslant i<j \leqslant k$, the starting point of car $j$ is higher than the starting point of car $i$; similarly, the finishing point of car $j$ is higher than the finishing point of car $i$. Say that two checkpoints are linked by some company if one can start from the lower checkpoint and reach the higher one by using one or more cars of that company (no movement on foot is allowed). Determine the smallest $k$ for which one can guarantee that there are two checkpoints that are linked by each of the two companies. Please provide the value of $k$ when $n=3$.
Answer: 7
Problem: Determine the largest $N$ for which there exists a table $T$ of integers with $N$ rows and 100 columns that has the following properties:
(i) Every row contains the numbers $1,2, \ldots, 100$ in some order.
(ii) For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r, c)-T(s, c)| \geqslant 2$. Here $T(r, c)$ means the number at the intersection of the row $r$ and the column $c$.The largest such integer is $N=a!/ 2^{b}$.求$a+b$.
Answer: 150
Problem: Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $\left(d_1, d_2, \ldots, d_k\right)$ such that for every $i=1,2, \ldots, k$, the number $d_1+\cdots+d_i$ is a perfect square. Please provide the sum of all such $n$.
Answer: 4
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.
Answer: 9
Problem: In each square of a garden shaped like a $2022\times2022$ board, there is initially a tree of height 0.A gardener and a lumberjack alternate turns playing the following game,with the gardener taking the first turn:

• The gardener chooses a square in the garden.Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. 
• The lumberjack then chooses four different squares on the board.Each tree of positive height on those squares then becomes one unit shorter.

We say that a tree is majestic if its height is at least $10^6$ .Determine the largest number $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board,no matter how the lumberjack plays.
Answer: 2271380
Problem: Let $n>3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbours on the right and left. Determine all initial distributions of coins from which it is possible that, after a finite number of steps, each child has exactly one coin. Please provide the value when $n=5$.
Answer: 15
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.
Answer: 1
Problem: Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_1, y_1\right)>f\left(x_2, y_2\right)$, we have $f\left(x_1+1, y_1\right)>f\left(x_2+1, y_2\right)$ and $f\left(x_1, y_1+1\right)>f\left(x_2, y_2+1\right)$.\n\nLet $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y<100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$. Please provide the sum of these two values.
Answer: 10000
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.
Answer: 16
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 $2^{n} - 1$.
Answer: 2025
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 $n = 6$.
Answer: 3
Problem: Let $n \geqslant 2$ be a positive integer. Paul has a $1 \times n^{2}$ rectangular strip consisting of $n^{2}$ unit squares, where the $i^{\text {th }}$ square is labelled with $i$ for all $1 \leqslant i \leqslant n^{2}$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or flipping) the pieces to obtain an $n \times n$ square satisfying the following property: if the unit square in the $i^{\text {th }}$ row and $j^{\text {th }}$ column is labelled with $a_{i j}$, then $a_{i j}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this. Please provide the value of the smallest number of pieces when $n=3$.
Answer: 5
Problem: The Imomi archipelago consists of $n \geqslant 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the traveller starts at, to visit all the islands exactly once (in particular, not returning to the island the traveller started at). Determine the maximal possible value of $k$ in terms of $n$. Please provide the value of $k$ when $n = 260$. The answer should be in the format of an integer.
Answer: 8
Problem: Determine all pairs $(a, p)$ of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all $a + p$ for the solutions.
Answer: 30
Problem: Let $\mathbb{Z}_{>0}$ be the set of positive integers. Determine all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$
f^{b f(a)}(a+1)=(a+1) f(b)
$$
holds for all $a, b \in \mathbb{Z}_{>0}$, where $f^{k}(n)=f(f(\cdots f(n) \cdots))$ denotes the composition of $f$ with itself $k$ times.Find the value of $f(5)$.
Answer: 4
Problem: At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac{7}{5}$ times as many matches as the girls. It is known that there were no draws. Find the smallest positive integer value of $n$ that satisfies the given condition.
Answer: 3
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 $a + b$ .
Answer: 5
Problem: Find all prime numbers $p, q, r$ such that $\frac{p}{q} - \frac{4}{r+1} = 1$. Please provide the sum of all  $p+q+r$
Answer: 37
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 $p$.
Answer: 12
Problem: Let $n>3$ be a positive integer. Equilateral triangle $ABC$ is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$. Please provide the value of $m-d$ when $n = 5$. The answer should be an integer.
Answer: 21
Problem: Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$. The original answer is in the format of an angle in degrees. Please provide the integer part of the degrees measure.
Answer: 45
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.
Answer: 14
Problem: The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. When \(a = 10\), find the volume of the pyramid. Round it to the nearest whole number.
Answer: 300
Problem: Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1,b_1,c_1,d_1$ of a geometric progression such that $a+a_1=27,b+b_1=27,c+c_1=39$, and $d+d_1=87.$ Please provide the sum of all terms in both progressions.
Answer: 180
Problem: The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\varphi$ between the cone's generating line and its base. Please provide the value of the parameter used in the arctan function, given that the original answer is in the format $\varphi=\arctan x$.Write down the value of $x$.
Answer: 4
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 $a=1$.
Answer: 12
Problem: Solve the system \(\left\{\begin{array}{l}x^3-y^3=2b,\\x^2y-xy^2=b.\end{array}\right.\) Determine the value of \(b\) for which there exists a solution \(x, y\).
Answer: 0
Problem: a) How many distinct ways are there of painting the faces of a cube six different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.) b) How many distinct ways are there of painting the faces of a dodecahedron with 12 different colors? (Colorations are considered distinct if they do not coincide when the dodecahedron is rotated.) Please provide the sum of the distinct ways for both a) and b).
Answer: 7983390
Problem: All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle such that the condition can be expressed as a simple integer calculation, e.g., height = base. Provide the sum of base and height if both are integers.
Answer: 10
Problem: What is the largest number of parts into which $n$ planes can divide space? Please provide the value of the expression when $n = 3$. The answer should be in the format \frac{m}{n}, directly give the value of m + n.
Answer: 9
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.
Answer: 9
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.
Answer: 179
Problem: Solve the equation: $$\begin{aligned}|x+1|-|x|+3|x-1|-2|x-2|=x+2.\end{aligned}$$ Please provide the sum of the minimum and maximum solutions.
Answer: 0
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.
Answer: 6
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.
Answer: 12
Problem: Divide $a^{2^k} − b^{2^k}$ by $(a + b)(a ^2 + b ^2 )(a ^4 + b^ 4 )\cdots (a^ {2 ^{k-1}} + b^ {2 ^{k-1}} )$. If $a=1$ and $b=-1$, what is the result of the division?
Answer: 1
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.
Answer: 503
Problem: Calculate (without calculators, tables, etc.) with accuracy to $0.00001$ the product \(\begin{pmatrix}1-\frac{1}{10}\end{pmatrix}\begin{pmatrix}1-\frac{1}{10^2}\end{pmatrix}\cdots\begin{pmatrix}1-\frac{1}{10^{99}}\end{pmatrix}.\) The original answer is a decimal number with five decimal places, please provide the value as an integer by removing the decimal point.
Answer: 89001
Problem: The sum of the reciprocals of three positive integers is equal to 1. What are all the possible such triples? Please provide the sum of all elements in these triples.
Answer: 30
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.
Answer: 12
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.
Answer: 239
Problem: 99 straight lines divide a plane into n parts. Find all possible values of n less than 199. Please provide the sum of all possible values.
Answer: 298
Problem: Delete 100 digits from the number $1234567891011 . . . 9899100$ so that the remaining number were as big as possible. Provide the sum of the remaining digits as an integer.
Answer: 409
Problem: Find all rectangles that can be cut into 13 equal squares. Provide the sum of the dimensions of the rectangle.
Answer: 14
Problem: What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point? Please provide the value of \binom{1954}{2}.
Answer: 1908081
Problem: Find all two-digit numbers x the sum of whose digits is the same as that of $2x, 3x,$ etc., $9x$. Please provide the sum of all such two-digit numbers.
Answer: 486
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.
Answer: 51
Problem: Find the number of integer values of n for which $N = 20^n + 16^n − 3 ^n − 1$ is divisible by 323.
Answer: 323
Problem: Solve the system: $$\begin{cases}\frac{2x_1^2}{1+x_1^2}=x_2,\\[2ex]\frac{2x_2^2}{1+x_2^2}=x_3,\\[2ex]\frac{2x_3^2}{1+x_3^2}=x_1.\end{cases}$$ Provide the sum of all possible values of x_1 + x_2 + x_3.
Answer: 3
Problem: Represent 1957 as the sum of 12 positive integer summands $a_1, a_2, . . . , a_{12}$ for which the number $a_1! · a_2! · a_3! · · · · · a_{12}!$ is minimal. Please provide the sum of all summands.
Answer: 327