MikaStars39/nano-eval huggingface dataset:
An explorer is challenging to pass through a maze composed of 3033 rows and 3032 columns of squares. The rules of the maze are as follows: In all the middle rows (that is, from the 2nd row to the 3032nd row), there is one invisible teleportation gate hidden in each row, and there are no multiple invisible teleportation gates in the same column. The explorer starts from any position in the 1st row. Each time, he can move one square up, down, left or right, and it is allowed to pass through the explored areas repeatedly. If the explorer touches a teleportation gate, he will immediately return to the starting point and start the next attempt. However, the explorer will mark the position of that teleportation gate. If the explorer reaches any position in the 3033rd row, he will directly pass the maze. It is known that the explorer can remember the positions of all the discovered teleportation gates. Please construct a general strategy so that no matter how the teleportation gates are distributed, the explorer will definitely pass the maze in the \(k\)th attempt in the worst-case scenario. Find the minimum possible value of \(k\).
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Answer: 3
For a positive integer \(n\geq2\)，define \(n - 1\) by \(n-1\) matrix \(M_n(\lambda)=\begin{pmatrix}6-\lambda&-\lambda&-\lambda&\cdots&-\lambda\\-\lambda&24-\lambda&-\lambda&\cdots&-\lambda\\-\lambda&-\lambda&60-\lambda&\cdots&-\lambda\\\vdots&\vdots&\vdots&\vdots&\vdots\\-\lambda&-\lambda&-\lambda&\cdots&n^{3}-n-\lambda\end{pmatrix}\). Let \(\lambda_n\)be the maximal \(\lambda\) such that \(\det(M_n(\lambda)) = 0\). Find \(I=\limsup_{n\rightarrow\infty}\lambda_n\).
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Answer: 4
A sequence of real numbers \(\{a_n\}\) satisfies that：\(a_{n + 1}=2^n-7a_n，n = 0,1,2,\cdots\). Find the minimal possible value of \(\frac{1}{a_0}\) such that \(a_{n + 1}>a_n\) for any positive integer \(n\).
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Answer: 9
A frog starts from 0 and makes 2047 jumps. This process satisfies the following conditions: (1) The frog will jump to each point in the set \(\{1, 2, 3,\cdots,2047\}\) exactly once without repetition or omission. (2) The length of each jump of the frog is a power of 2, and the direction can be either to the left or to the right. Let \(T\) be the sum of the reciprocals of the lengths of the frog's jumps. Find the minimum value of \(T\).
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Answer: 11
For any positive integer \(n\geq2\), we define \(f(n)\) as the last non-zero digit of \(n!\) in the decimal system. It is known that \(k\) is the fifth smallest positive integer that satisfies the following condition: for any positive integer \(n\geq2\), \(f(kn) = f(n)\) always holds. How many digits does \(k\) have in the decimal system?
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Answer: 12
How many integers \(k\in[2,18676]\) are there such that the number of positive integer triples \((n,a,b)\) satisfying the following conditions is finite: \(n\) is odd, \(\gcd(a,b) = 1\), \(a + b=k\), and \(n\mid a^{n}+b^{n}\)?
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Answer: 14
There are 60 real numbers \(a_1, a_2, \cdots, a_{60}\) satisfying \(a_1 + a_2+\cdots + a_{60}=0\), and for any \(1\leqslant i\leqslant 60\), \(\vert a_i - a_{i + 1}\vert\leqslant 1\) (\(a_{61}=a_1\)). If \(a = a_{12}\), \(b = a_{24}\), \(c = a_{36}\), \(d = a_{48}\), \(e = a_{60}\). Find the maximum value of \(a + b + c + d+e\) .
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Answer: 15
At least how many black stones should be placed on a 9×9 square grid board to ensure that there is no possibility for the white to make a five-in-a-row? (which means there are no five consecutive empty squares in a row, in a column, or five consecutive empty squares along a diagonal among the remaining empty squares). Note that empty square here means a square with no black stone in it and each square can only be filled with no more than 1 black stone.
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Answer: 16
\(n\) students take an exam. There are a total of 6 questions in the exam, and each question has three options. After the exam, it is statistically found that for any two students, the number of questions with the same answers between them is either 0 or 2. What is the maximum value of \(n\)?
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Answer: 18
Place \(m\) \(1 \times 2\) dominos on a \(7\times7\) chessboard such that no domino can slide horizontally or vertically when other dominos remains fixed. Find the minimal possible value of \(m\).
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Answer: 22
Let \(N\) be the minimal positive integer which can not be represented as \(\sum_{i = 1}^{7263}(- 1)^{a_{i}}\times2^{b_{i}}\) where \(a_{i},b_{i}(i = 1,2,\cdots,7263)\) are non-negative integers. Find \(N \mod 29\).
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Answer: 23
How many squares at least should be marked on a \(13\times13\) chessboard such that for any placement of a bishop on the chessboard, the bishop can threaten at least one of the marked squares? (A bishop can threaten the squares on the same diagonal line, and if a bishop is placed on a marked square, it is also considered that the square is threatened.)
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Answer: 24
Find the largest positive number \(C\)，such that for any six positive integers \(x_1, x_2, \ldots, x_6\)，we have\( \left(x_1+\cdots+x_6\right)^3  \geq C \operatorname{gcd}\left(x_1, x_2, \ldots, x_6\right) \cdot \sum_{i=1}^6 \left(\operatorname{gcd}\left(x_i, x_{i+1}\right) \cdot \operatorname{lcm}\left(x_{i+1}, x_{i+2}\right)\right) \)
where \(x_{7}=x_1, x_{8}=x_2\).
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Answer: 27
Place 471 coins in the squares of an \(n\times n\) board where each block can hold 1 coin, multiple coins, or no coin at all, such that the difference in the number of coins between any two adjacent squares (squares that share a common side) is 1. Find the maximum value of \(n\) for which such an arrangement is possible.
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Answer: 29
For integers \(a_1,a_2,\cdots,a_{7926}\in\{1,2,\cdots,500\}\), denote \(S = \sum_{i = 1}^{7926}\left[\frac{a_{i + 1}}{a_i}\right]\), where \(a_{7927}=a_1\). Find the minimal possible value of \(S\).
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Answer: 32
Find the maximum number of "kings" that can be placed on a \(10\times10\) chessboard, such that each "king" can exactly attack ONE other "king" (a "king" can attack the chess piece in the square that shares a common vertex with the square it is in).
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Answer: 40
Kevin throws three fair dice numbered from 1 to 6. Then he observes the results of the first throw of the three dice. From these three dice, he can choose \(n\), where \(0\leq n\leq3\) dice to re-throw once. After that, the operation ends. Kevin's goal is to maximize the probability that the sum of the numbers on the final three dice is 7, and Kevin is smart enough. Let the probability when \(n = 2\) be \(p\). What is the value of \(216\times p\)?
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Answer: 42
Let \(m\) be the maximal positive integer such that there exists complex numbers \(r_1,\cdots,r_{645}\), which are not all zeros, satisfying \(\prod_{k = 1}^{645}(r_k + 1)=\prod_{k = 1}^{645}(r_k^2+1)=\cdots=\prod_{k = 1}^{645}(r_k^m + 1)=1\). Find \(m \mod 848\).
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Answer: 46
Given a \(15\times15\) chessboard, there are \(n\) pieces placed on \(n\) individual squares respectively. For each piece, a \(5\times5\) square with its sides parallel to the boundaries of the chessboard is drawn with the piece located as the center, and this square is called the range of the piece. If there is exactly one other piece within the range of each piece, find the maximum value of \(n\).
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Answer: 48
Let \(\left\{a_n\right\}\) satisfies \(a_1=\frac{2}{5}, a_n^2=a_{n+1}-2 a_n^2 a_{n+1}\), find the residue of \(\left[\frac{1}{a_{2025}}\right]\) mod 407.
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Answer: 53
It is known that one plane divides the space (three-dimensional) into two parts, two parallel planes divide the space into three parts, and two intersecting planes divide the space into four parts; Consider the planes where the six faces of the cube \(ABCD - A_1B_1C_1D_1\) are located, and the planes where the four faces of the tetrahedron \(BA_1C_1D_1\) are located. How many parts does these ten planes divide the space into?
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Answer: 64
Let \(S = \{\frac{2y - 3}{2x - 1} \mid \min\{\vert x - 2\vert,\vert x + 2\vert\}+\vert y\vert\leqslant1\}\), Assume the length of the set \(S = \frac{m}{p}\) where \(m\) and \(p\) are coprimes, Find \(m+p\).
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Answer: 67
Given that the quadrilateral inscribed in the sphere is an isosceles trapezoid \(ABCD\), where \(AB = 2\), \(BC = AD = 3\), \(CD = 4\), and the center of the sphere is \(O\). Points \(E\) and \(F\) are respectively on the sides \(AD\) and \(BC\), and the minimum value of \(\overrightarrow{OE}\cdot\overrightarrow{OF}\) is \(- 2\). Find the minimum interger which is greater than the volume of the sphere.
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Answer: 70
Suppose there is a necklace composed of beads of two colors, red and blue, with a total of 16 beads. And the necklace can be freely rotated and flipped (that is, the cases regarded as the same necklace need to be combined for counting). If on this necklace, there are neither three consecutive blue beads nor three consecutive red beads, how many different necklaces that meet the above requirements are there?
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Answer: 88
Let the plane be tiled with an infinite number of unit squares arranged in a checkerboard pattern. If another unit square falls randomly on this plane, and its position and orientation are independent of the tiling of the checkerboard, let \(p\) be the probability that this unit square does not cover any vertex of the squares in the checkerboard. Find \(\lfloor 1000\cdot p\rfloor\) (the result of rounding down the product of \(1000\) and \(p\)).
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Answer: 90
Real numbers \(x_1 \leq x_2 \leq \ldots \leq x_9\) such that \(\sum_{i=1}^9 x_i=1\). Find the maximum value of 
\(
\frac{\left(\sum_{i=1}^9\left|x_i-x_{5}\right|\right)^2 \cdot \sum_{1 \leq i<j \leq 9}\left(x_i-x_j\right)^2}{\left(\sum_{i=1}^9 x_i^2\right)^3}
\).
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Answer: 96
In a square grid table composed of \(16\times16\) unit squares with side length \(1\), \(n\) squares are selected. In each of the selected squares, a directed diagonal is drawn. It is known that for any two directed diagonals, either the end point of one diagonal is the same as the starting point of the other diagonal, or the distance between the two end points is at least \(2\). Find the maximum value of \(n\).
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Answer: 108
A set of 2025 squares in a \(2025\times2048\) chessboard is called a generalized diagonal (simply called a diagonal for short) if any two of these squares are neither in the same row nor in the same column. Some squares in the \(2025\times2048\) chessboard are colored red such that there are at most 128 red squares in each row and each column. Try to find the minimum value of the positive integer \(n\) such that for any coloring method, we can always find \(n\) diagonals on the chessboard, such that all the red squares are on these diagonals.
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Answer: 128
Let \(a_1,a_2,\cdots\) be an infinite sequence of positive integers with \(a_1 = 1\), such that\(a_n\mid a_k + a_{k + 1}+\cdots + a_{k + n - 1}\)  for all positive integers \(k\) and \(n\). Let \(N\) be the maximal possible value of \(a_{5652}\), find the \(N \mod 776\).
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Answer: 143
If integer \(n\) satisfies \(a_1=n^2-10 n+23, a_2=n^2-9 n+97, a_3=n^2-12 n+46\) are all primes, Find the sum of all possible value of \(a_1+a_2+a_3\).
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Answer: 196
The 30 edges of a regular icosahedron are distinguished by labeling them 1, 2, ..., 30. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? Find the answer mod 1000.
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Answer: 224
Assume a positive integer \(N\) can be represented as summation of \(k\) consecutive positive integers, i.e. \(N = a + (a + 1) + (a + 2) + \cdots + (a + k - 1)\), for \(k = 1559\), and can not be represented as summation of \(k\) consecutive positive integers for any other \(k > 1\). Consider all positive integers \(N\) with the aforementioned property, find the minimal positive integer \(a\) in all of the representations of all \(N\).
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Answer: 245
Real numbers \(a,b,c>0\) where \(a+b+c=3\), let the maximal value of \((a^3+b^3+c^3)*(a+b)*(b+c)*(c+a)\) be \(p = \frac{m}{n}\) where \(m\) and \(n\) are coprimes. Find \(m+n\).
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Answer: 247
Ten points are randomly and independently generated with a uniform distribution inside a circle. Denote the probability that the center of the circle lies outside the convex hull of these ten points as \(p = \frac{m}{n}\), where \(m\) and \(n\) are coprime. Find \(n - m\).
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Answer: 251
A coin is randomly drawn from a population. The probability that this coin shows heads when tossed once is \(p\), and \(p\) follows a uniform distribution \(U[0,1]\). Subsequently, this randomly drawn coin is tossed repeatedly for a total of 256 times. Let \(p = \frac{m}{n}\) be probability of getting exactly 128 heads where \(m\) and \(n\) are positive coprime integers. Find \(m+n\).
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Answer: 258
There is a function \( g: \mathbb{N} \times \mathbb{N} \to \mathbb{N} \) such that \( g(0,0) = 0 \), and for any \( x, y \in \mathbb{N} \), there exists \( n \in \mathbb{N} \) such that
\[\{g(x,y), g(x,y+1), g(x+1,y)\} = \{n, n+1, n+2\}.\]
Let \(S\) be the set of all possible values of \( g(4000, 4036) \). Find the 100-th smallest value of \(S\).
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Answer: 297
Nine balls numbered \(1, 2,\cdots,9\) are randomly placed on the nine equally-divided points of a circle, with one ball at each point. Let \(S\) be the sum of the absolute values of the differences between the numbers of all adjacent pairs of balls on the circle. Find the reciprocal of the probability of the placement method that makes \(S\) reach the minimum value. (Note: If a certain placement method can coincide with another placement method after rotation or mirror reflection, they are considered the same placement method.)
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Answer: 315
For a positive integer \(n\), sort all the \(l\) positive factors of \(n\) as \(d_1, d_2, \cdots, d_l\) from smallest to largest. If there exist positive integers \(i, j \in\{1,2, \cdots, l\}\), such that \(d_8^2+d_i^2=d_j^2\), find the minimal value of \(n\)．
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Answer: 360
\(m\) distinct positive even numbers and \(n\) distinct positive odd numbers sum up to 2025. Find the maximal value of \(5m+7n\) among all such \(m\) and \(n\).
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Answer: 384
Bob has a convex 735-gon in the plane and
wishes to construct the centroid of its vertices. He has no standard ruler or compass, but
he does have a device with which he can dissect the segment between two given points
into m equal parts. For how many \(m \in [9, 81256]\) can Bob necessarily accomplish his task?
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Answer: 386
For two sets \(A\) and \(B\), the set \(A\Delta B=\{x|x\in A\text{ and }x\notin B\}\cup\{x|x\in B\text{ and }x\notin A\}\) is defined. Let \(A_1,A_2,\cdots,A_{2^{312}}\) be all the subsets of \(S = \{1,2,\cdots,312\}\), and \(M\) be a number table with \(2^{312}\) rows and \(2^{312}\) columns. For any \(i,j\in\{1,2,\cdots,2^{312}\}\), the sum of all the elements in the set \(A_i\Delta A_j\) (it is agreed that the sum of the elements of the empty set is \(0\)) is filled in the position of the \(i\)-th row and \(j\)-th column of the number table \(M\). Find the remainder when the sum of all the numbers in the number table \(M\) is divided by \(1000\).
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Answer: 424
Real numbers \(x_1,x_2,\cdots,x_n\) satisfy \(|x_1|+|x_2|+\cdots+|x_{3526}| = 1\). Let \(M\) be the maximal possible value of \(\frac{1}{|x_1|+|x_1 - x_2|+|x_1 + x_2 - x_3|+\cdots+|x_1 + x_2+\cdots+x_{3525}-x_{3526}|}\). Find \(M \mod 1000\).
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Answer: 432
Given \(x_1^2+y_1^2+z_1^2=1,6 x_2+3 y_2+2 z_2=12\), where \(x_1, y_1, z_1 \in \mathbb{R}, x_2, y_2, z_2 \in(0,+\infty)\). Let the minimal value of \(\max \left\{\left|x_2-x_1\right|,\left|y_2-y_1\right|,\left|z_2-z_1\right|\right\}\) be \(x\), Find the largest integer which is smaller than \(1000 \cdot x\).
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Answer: 454
A \(12\times12\) matrix \(A\) satisfies that each row has exactly \(6\) elements equal to \(1\) and \(6\) elements equal to \(0\). Then, how many matrices \(A\) are there such that \(A^{2}+6A\) is a matrix with all elements equal to \(6\)?
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Answer: 462
Let \(\mathbb{F}_{2027}\) be the field of residues modulo \(2027\). Let \(W\) be the smallest set of polynomials with coefficients in \(\mathbb{F}_{2027}\) such that the polynomials \(x+1\) and \(x^{2025}+x^{2024}+\cdots+x^2+2x+1\) are in \(W\), and for any polynomials \(h_1(x)\) and \(h_2(x)\) in \(W\), the polynomial \(r(x)\), which is the remainder of \(h_1(h_2(x))\) modulo \(x^p-x\) is also in \(W\). Let \(N\) be the number of polynomials in \(W\). How many consecutive zeros are there at the end of the decimal representation of \(N\)?
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Answer: 505
For every real number \(x\), let \(f(x)=\sum_{n\in S_x}\frac{1}{2^n}\), where \(S_x\) is the set of positive integers \(n\) such that\(\lfloor nx\rfloor\) is even. Let\(L_{max}\) be the largest real number such that \(f(x)\geq L\) for all \(x\in[0, 1)\). Find the largest integer which is smaller than \(1000 \cdot L_{max}\).
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Answer: 571
Consider a pair of integers \(a, b\) such as \(a^2+b^2 \leq 19\), and there exists real numbers \(x, y\) such that \(\frac{a}{\sqrt{x}}+\frac{2}{y+3 x}=\frac{b}{\sqrt{y}}-\frac{2}{y+3 x}=1\)．Find the square of the numbers of ordered pair \((a, b)\) satisfying the aforementioned property.
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Answer: 576
Find the minimal number \(C\) satisfying：
\[
\sum_{0 \leq i,j \leq n} 4^{i+j+1} C_{i+j}^i x_i x_j+\sum_{0 \leq i \leq n} x_i \geq -\frac{1}{4144}C 
\]
for all positive integers \(n\) and \(x_0, x_1, \cdots, x_n \in \mathbb{R}\).
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Answer: 592
Consider positive integers \(a_1<a_2<\cdots<a_{38}\) satisfying \(a_1+a_2+\cdots+a_{38}=1997\). Let \(b_k=\min \left\{a_1+a_2+\cdots+a_k, a_{k+1}+a_{k+2}+\cdots+a_{38}\right\}\). Let \(S=\max \left\{b_1, b_2, \cdots, b_{37}\right\}\). Find the minimal value of \(S\)．
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Answer: 666
Arrange the numbers \(1, 2,\dots,333^{2}\) into a \(333\times333\) number table, with each number appearing exactly once. Calculate the ratio of the larger number to the smaller number for every pair of numbers in the same row or the same column. The smallest value among these \(333^{2}\times332\) ratios is called the characteristic value of this number table. Let the maximum possible value of the characteristic value be \(\lambda=\frac{\alpha}{\beta}\), where \(\alpha\) and \(\beta\) are coprime. Find \(\alpha + \beta\).
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Answer: 667
Let the radius of the circle \(O\) be \(r = 1\), and \(M\) be a point on the edges of regular dodecagon (12-sided polygon) inscribed in the circle \(O\). \(E\) and \(F\) are a pair of diametrically opposite points on the circle \(O\). Let \(S=\min\{\overrightarrow{MF}\cdot\overrightarrow{ME}\}\). Find the value of \(\lceil- 10000\times S\rceil\).
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Answer: 670
For a finite non-empty set \(X\) of real numbers, denote the number of elements in \(X\) as \(\vert X\vert\), and \(f(X)=\frac{1}{\vert X\vert}(\sum_{a\in X}a)\). The pair of sets \((A,B)\) satisfies \(A\cup B = \{1, 2, \cdots, 200\}\), \(A\cap B=\varnothing\), and \(1\leq\vert A\vert\leq198\) and \(2\leq\vert B\vert\leq199\). For any \(p\in B\), let \(A_p = A\cup\{p\}\) and \(B_p=\{x\vert x\in B,x\neq p\}\). For all \((A,B)\) that satisfy the above conditions and \(p\in B\), find the maximum value of \((f(A_p)-f(A))\cdot(f(B_p) - f(B))\). If it is in fraction form, calculate the sum of the numerator and the denominator of the irreducible fraction.
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Answer: 724
There is a sphere with radius \(\sqrt{6}/36\) moving freely inside a tetrahedron with side length 1. Let \(V\) be the volume of the reachable regions of the center of the sphere. Find  \(\lfloor 1/V^2 \rfloor\).
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Answer: 820
Find the minimal positive integer \(n \geqslant 866\) satisfying: there exists a bijection \(f\) from \(\{1,\cdots,n\}\) to itself, such that for any \(x,y\in\{1,\cdots,n\}\), if \(n\) divides \(866x - y\) then \((n + 1)\mid f(x)^{866} - f(y)\).
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Answer: 876
Let \(x_i=\frac{i}{1771}(i=0,1, \cdots, 1771)\), \(S=\sum_{i=0}^{1771} \frac{x_i^5}{5 x_i^4-10 x_i^3+10 x_i^2-5 x_i+1}\), Find \(S\).
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Answer: 886
Given non-negative real numbers \(x, y, z\) satisfying \(\sqrt{x^2+y^2}+z=1\), for positive real number \(k\), define \(f(k)\) as the maximal value of \(4(xy+kxz)\). Find the largest integer no more than \(f^2(1)+3f^2(2)+8f^2(3)+15f^2(4)+24f^2(5)\).
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Answer: 979
Let \(I_{n}=\int_{0}^{\frac{\pi}{2}}\frac{\sin^{2}(nt)}{\sin t}dt\), find the largest integer smaller than \(1000 \times \lim_{n\rightarrow\infty}\frac{I_{n}}{\log_10{n}}\).
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Answer: 1151
Find the minimal constant \(k\), such that for any positive real numbers \(x,y,z\), the inequality \(\frac{x}{\sqrt{x+y}}+\frac{y}{\sqrt{y+z}}+\frac{z}{\sqrt{z+x}}\leq \frac{k}{1000}\sqrt{x+y+z}\) always hold true.
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Answer: 1250
Given 1893 people and 66 kinds of shapes. Each person holds one cookie of each of the 66 shapes, and the total mass of these 66 cookies is 1 (the mass of each cookie is a positive real number; the masses of cookies of the same shape held by different people can be different). Let \(C\) be the smallest positive real number such that it is always possible to select one cookie from each person, satisfying that for each kind of shape among the selected cookies, the total mass of cookies of this shape does not exceed \(C\). Suppose \(C\) is written in the form of the simplest fraction \(\frac{q}{p}\), find the value of \(p + q\).
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Answer: 1403
Let \(S = \{1,2,\cdots,6172\}\) and \(M\) is the number of functions \(f:S\rightarrow S\) that satisfy the following conditions: 
(1) \(f\) is a one-to-one mapping. 
(2) For any positive integer \(a\), (\(1\leq a\leq6172\)), \(f(a)+f^{-1}(a)=6173\) holds, where \(f^{-1}\) is the inverse function of \(f\). 

Find the value of \(k\) where \(M=2^{k}\).
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Answer: 1543
Find the smallest positive integer \(t\), such that for any real numbers \(a_1, a_2, \cdots, a_{4467}\), the inequality \(\max(|a_1|, |a_2|, \cdots, |a_{4467}|) \leq t \cdot \max(|a_1 + a_2|, |a_2 + a_3|, \cdots, |a_{4466} + a_{4467}|, |a_{4467} + a_1|)\) always hold true.
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Answer: 2234
\(x_1,x_2,\dots,x_{2025}\) are positive real numbers. Let \(M\) be the minimal value of \(\max\{x_1,\frac{1}{x_1}+x_2,\dots,\frac{1}{x_{2024}}+x_{2025},\frac{1}{x_{2025}}+1\}\). Suppose the a polynomial \(F(x)\) with integer coefficients can be divided by \(x^2 - M x + 1\), find the minimal degree of \(F(x)\).
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Answer: 2304
In a \(60\times60\) square grid of cells, the integers \(1, 2,\cdots,3600\) are filled in such a way that the sum of the numbers in every two adjacent cells (cells that share a common side are called adjacent) is not less than \(S\). Try to find the maximum value of \(S\).
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Answer: 3571
Let non-real complex number \(z\) satisfying \(z^{4051}+z^{4050}+5=0\). Find the number of such \(z\)'s (multiplicity are counted)．
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Answer: 4050
Let \(P(x),Q(x)\) be distinct 4046-degree real polynomials with non-zero coefficients. Let \(r\) be the number of their common real roots (multiplicity counted) and \(s\) be the number of their common terms. Find the maximum possible value of \(r + s\).
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Answer: 6068
A regular octagon is divided into 8 regions, namely \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), and \(H\) by connecting the center point with each vertex. Now we place one ball in each region. There are a total of four different colors of balls, and there are infinitely many balls of each color. It is required that balls of the same color cannot be placed in adjacent regions. Find out how many ways there are to place the balls in total. Note that the rotationally or reflectively symmetric ways are NOT considered equivalent.
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Answer: 6564
There are 100 observers inside a finite circular playground (the positions of any two observers are different). Each observer has a viewing angle of \(100^{\circ}\) and can observe all the other observers within its viewing angle range (including those on the two sides of the angle) inside the playground. (Each observer is regarded as a point, and these 100 observers are in the same plane.) For \(i = 1, 2,\cdots,100\), let \(a_i\) be the number of all the observers other than itself that the \(i\)-th observer can observe. Find the maximum value of \(\sum_{i = 1}^{100}a_i\).
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Answer: 8700
Denote \(x\succ y\) as \(0\lt x - y\leqslant9\) or \(y - x\gt9\). Let \(a,b,c,d\in\{1,2,\cdots,19\}\). For the cases where \(a\succ b\), \(b\succ c\), \(c\succ d\) and \(d\succ a\), find out how many such four-tuples \((a,b,c,d)\) there are.
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Answer: 9120
Let \(x_k\in [-2,2],k=1,2\dots,4698\) satisfying \(x_1+x_2+\dots +x_{4698}=0\). Find the maximal value of \(M=x_1^3+x_2^3+\dots +x_{4698}^3\).
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Answer: 9396
The rectangle \(R\) is divided into 2024 small rectangles, and the sides of each small rectangle are parallel to the sides of \(R\). The vertices of the small rectangles are called nodes. A line segment on the sides of the small rectangles, if both of its endpoints are nodes and there are no other nodes inside it, is called a fundamental line segment. Considering all the ways of division, find the sum of the maximum value and the minimum value of the number \(N\) of fundamental line segments.
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Answer: 10211
Let \(G\) be a 160-order simple graph. For any vertex \(u\), there exists another vertex \(v\), such that \(u\) and \(v\) are adjacent, and there exists no vertices adjacent to both \(u\) and \(v\). Find the maximal number of edges in \(G\).
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Answer: 10320
Choose a subset of integers from 1 to 15 such that no three consecutive integers are in the chosen set (you can choose an empty set). How many different sets are there?
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Answer: 10609
Let \(\mathbb{Z} / n\mathbb{Z}\) be the set of all integers modulo \(n\) (hence \(\mathbb{Z} / n\mathbb{Z}\) contains \(n\) elements). Find the second smallest composite number \(n\), such that there exists a bijective function \(g:\mathbb{Z} / n\mathbb{Z} \to \mathbb{Z} / n\mathbb{Z}\) such that the \(101\) functions \(g(x), g(x)+x, g(x)+2x, \cdots, g(x)+100x\) are all bijective from \(\mathbb{Z} / n\mathbb{Z}\) to \(\mathbb{Z} / n\mathbb{Z}\).
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Answer: 11021
The Bank of Pittsburgh issues coins that have a heads side and a tails side. Vera
has a row of 7873 such coins alternately tails-up and heads-up, with the leftmost
coin tails-up.
In a move, Vera may flip over one of the coins in the row, subject to the following
rules:
• On the first move, Vera may flip over any of the 7873 coins.
• On all subsequent moves, Vera may only flip over a coin adjacent to the coin
she flipped on the previous move. (We do not consider a coin to be adjacent
to itself.)
Determine the smallest possible number of moves Vera can make to reach a state in
which every coin is heads-up.
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Answer: 15744
Among all the lattice points \((x, y)\) where \(1\leqslant x\leqslant9973\) and \(1\leqslant y\leqslant9973\), some lattice points are to be selected. The restrictive condition is that among the selected lattice points, no four points can form an isosceles trapezoid (a rectangle is also regarded as an isosceles trapezoid) with the base parallel to either the \(x\)-axis or the \(y\)-axis. So, what is the maximum number of lattice points that can be selected?
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Answer: 24931
Each edge of the complete graph \(K_{9097}\) is labeled with \(1\), \(2\), or \(3\) such that the sum of the labels of any triangle (formed by three edges) is at least \(5\). Given that the minimum possible value of the average of all \(\binom{9097}{2}\) labels can be expressed as a fraction in its simplest form \(\frac{m}{n}\), find the value of \(m + n\).
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Answer: 27290
Integers \(a_1,a_2,\cdots,a_{2025}\) satisfy \(1 = a_1\leq a_2\leq\cdots\leq a_{2025}=199\). Let \(f=(a_1^2 + a_2^2 + \cdots + a_{2025}^2)-(a_1a_3 + a_2a_4 + a_3a_5 + \cdots + a_{2023}a_{2025})\). Find the minimal value of \(f\).
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Answer: 29800
Let \(S\) be the set of integers \(s\) satisfying \(s\geq3456\), and there exists positive integers \(a\), \(b\), \(c\), \(d\) such that \(s = a + b + c + d\) and \(s\) divides \(abc + abd + acd + bcd\). Find the sum of ten smallest elements in (S\).
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Answer: 34628
The number of ways to color an \(n \times n\) chessboard with black and white such that no neighboring two blocks can be black simultaneously is \(x_n\). Find \(\sum_{n=2}^5 x_n\)．
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Answer: 56751
The values of the 20 numbers \(a_1, a_2, \cdots, a_{20}\) can be either \(0\) or \(1\). If two adjacent terms \(a_k\) and \(a_{k + 1}\) are \(1\) and \(0\) respectively, then it is called a "reverse order". So, how many sequences are there that have exactly 3 "reverse orders"?
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Answer: 116280
Real numbers \(x_{1},x_{2},\cdots,x_{1997}\) satisfy the following two conditions: (1)\(-\frac{1}{\sqrt{3}}\leq x_{i}\leq\sqrt{3}\)（\(i = 1,2,\cdots,1997\); (2)\(x_{1}+x_{2}+\cdots +x_{1997}=-318\sqrt{3}\). 
Let \(M\) be the maximal possible value of \(x_{1}^{12}+x_{2}^{12}+\cdots +x_{1997}^{12}\). Find the largest integer no more than \(M\).
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Answer: 189548
Let \(a_1,a_2,...,a_m\) be \(m\) distince positive integers such that \(a_1\cdot a_2\cdot...\cdot a_m\) has 606 prime factors. If the product of any number of terms among \(a_1, a_2, \cdots, a_m\) is not the 607th power of some integer, find the maximal value of \(m\) in an integer form.
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Answer: 367236
There are 1038 equally spaced points on a circular track \(\gamma\) of circumference 1038. The points are labeled \(A_1, A_2, \dots, A_{1038}\) in some order, each label used once. Initially, Bunbun the Bunny begins at \(A_1\). She hops along \(\gamma\) from \(A_1\) to \(A_2\), then from \(A_2\) to \(A_3\), until she reaches \(A_{1038}\), after which she hops back to \(A_1\). When hopping from \(P\) to \(Q\), she always hops along the shorter of the two arcs \(\hat{PQ}\) of \(\gamma\); if \(\bar{PQ}\) is a diameter of \(\gamma\), she moves along either semicircle. Determine the maximal possible sum of the lengths of the 1038 arcs which Bunbun traveled, over all possible labellings of the 1038 points.
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Answer: 537686
Consider pairs \((f, g)\) of functions from the set of nonnegative integers to itself such that \(f(0) \ge f(1) \geq f(2) \geq \cdots \ge f(2016) \ge 0\) and \(f(0)+f(1)+f(2)+\cdots +f(2016) \le 2016\). For any 20 nonnegative integers \(n_{1}, n_{2}, \dots, n_{20}\), not necessarily distinct, we have \[
g(n_{1}+n_{2}+\cdots+n_{20}) \le f(n_{1})+f(n_{2})+\cdots+f(n_{20})
\] 
Determine the maximum possible value of \(g(0)+g(1)+\cdots+g(2016)\) over all such pairs of functions.
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Answer: 794430
Let \(G\) be a \(499\times3997\) grid table, and a number is filled in each cell of \(G\). A rectangular sub-grid table \(S\) of \(G\) is called "outstanding" if the number in each cell of \(S\) is greater than the number in each cell outside \(S\) that shares a common vertex with \(S\). The grid table \(G\) itself is also an outstanding sub-grid table. Find the maximum value of the number of outstanding sub-grid tables.
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Answer: 999499
Given positive integer \(n \geq 3\), \(x_1, x_2, \cdots, x_n \in[0,1]\), consider the set \(A=\left\{(i, j)\left|1 \leq i<j \leq n, i, j \in \mathbb{N}^*, i\right| j\right\}\), let \(d(n)\) be the maximum value of \(x_1^4+x_2^4+\cdots+x_n^4-\sum_{(i, j) \in A} x_i^3 x_j^3\). Find \(\sum_{k=3}^{2025} d(k)\).
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Answer: 1026167
Consider a regular 25-prism. By connecting the face diagonals of each lateral face, 25 intersection points are generated, and then the lateral edges are removed. Now, we perform a three-coloring on these 25 intersection points and 50 vertices, where adjacent vertices cannot have the same color. Colorings that coincide after rotation around the central axis are regarded as the same kind. Find the number of coloring methods.
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Answer: 1342182
If there exist positive integers \(a\) and \(b\) (\(a < b\)) such that \(a^{3}=(ka + 1)b + k\), then the integer \(k\) is called "good". Then, find the sum of all "good" integers \(k\) that do not exceed 1997.
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Answer: 1995002
Kumar has a regular 2024-gon, and there is a coin on each vertex of it. In one operation, he selects a vertex, takes one coin from each of the two adjacent vertices to this selected vertex, adds one coin to the selected vertex, and keeps the extra coin for himself. He can perform such an operation only when both of the two adjacent vertices have at least one coin. He stops when he can no longer perform any operation. Let the maximum value of the number of coins he can collect be \(N_{max}\), and the minimum value be \(N_{min}\). Find the value of \(1000\cdot N_{max}+N_{min}\).
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Answer: 2022676
On a \(2025\times2025\) grid table, one white piece and one black piece are randomly placed in different cells. Alice moves the white piece first, then Bob moves the black piece, and they move alternately, and the following rules are satisfied: Each move can only be one cell horizontally or vertically, and for the same piece, horizontal and vertical moves must be alternated. If after Bob's move, the black piece and the white piece exactly fall into the same cell, then Bob wins; otherwise, Alice wins. Let \(p = m/n\) be the probability that Bob has a winning strategy where \(m\) and \(n\) are coprime. Find \(m\).
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Answer: 2050312
The side lengths of a convex 2025-gon are \(x_1, x_2, \cdots, x_n\), where \(x_i\in\mathbb{Z}_{+}\) (\(1\leq i\leq n\)). Let \(f(m)\) denote the minimum value of \(x_1x_2\cdots x_n\) when the perimeter of this convex polygon is \(m\). Find the value of \(f(2025) + f(2026)+\cdots + f(4050)\).
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Answer: 2061442
At a party, the host wants to distribute 2025 pieces of cookies of the same size to \(n\) guests and follow the following two rules: (1) Each cookie can be divided into at most two parts (does not have to be equally divided). (2) The total amount of cookies received by each guest is equal. Find the sum of all positive integers \(n\) for which such a distribution method exists.
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Answer: 2085451
In an \(n\times n\) (\(n\in\mathbb{N}^*, n\geq2\)) chessboard, each unit square is filled with the number \(0\). In each operation, two unit squares with a common side are selected, and the numbers in them are both increased by \(1\). If it is possible, through several (more than \(0\) times) operations, to make the sum of the numbers in each row and each column be the same value, then the minimum number of operations is denoted as \(f(n)\); if there is no such operation, then \(f(n) = 0\). Find \(\sum_{i = 2}^{2025}f(i)\).
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Answer: 3077492
For positive integer \(n\) and real numbers \(a_1, a_2, \ldots, a_n \in[-1,1]\). Let \(S=\sum_{i=1}^n a_i\) and \(f(n)\) be the maximum value of \(\sum_{i=1}^n\left|\frac{S-a_i}{n}-a_i\right|\). Find \(\sum_{k=1}^{225} kf(k)\)．
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Answer: 3847537
Non negative real numbers \(x_1, x_2, \cdots, x_{2025}\) sum up to 1. Let \(S = \frac{m}{n}\) be the maximal value of \(\sum_{1 \leq i, j \leq 2025} \frac{2 i-1}{2 j-1} x_i x_j\) where \(m\) and \(n\) are coprime, find \(m+n\).
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Answer: 4104674
There are 13 cards, on which the positive integers \(2,3,\cdots,14\) are written respectively. These cards are randomly arranged in a row from left to right, and the number on the \(i\)-th card is denoted as \(a_i\), where \(i = 1,2,\cdots,13\). Let the probability that \(a_i\geqslant i\) for all \(i = 1,2,\cdots,13\) be \(\frac{a}{x}\), where \(a\) and \(x\) are relatively prime positive integers. Find the value of \(a + x\).
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Answer: 6081079
Let \(A = \{1,2,\cdots,12\}\), and \(f\) be a one-to-one mapping from \(A\) onto \(A\). Denote \(f_1(x)=f(x)\) and \(f_{n + 1}(x)=f(f_n(x))\). The mapping \(f\) satisfies the following conditions: (1) For all \(x\in A\), \(f(x)\neq x\); (2) For all \(x\in A\), \(f_{35}(x)=x\). Question: How many mappings \(f\) that satisfy the above conditions are there?
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Answer: 13685760
There are 7671 piles of coins, denoted as \(S_1, S_2, \cdots, S_{7671}\) respectively. Initially, for each \(k = 1, 2, \cdots, 7671\), there are \(k\) coins in the pile \(S_k\). In one operation, an ordered pair \((i, j)\) is selected, where \(i\) and \(j\) satisfy the following two conditions: - \(1\leq i < j\leq7671\); - There is at least one coin in both the pile \(S_i\) and the pile \(S_j\); - The ordered pair \((i, j)\) has not been selected in any previous operation. Then, if there are \(a\) coins and \(b\) coins in \(S_i\) and \(S_j\) respectively, \(\gcd(a,b)\) coins are removed from each of the two piles. What is the maximum number of times this operation can be performed?
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Answer: 14711060
Take out 17 balls from an ample supply of red, green, and blue balls (these balls are completely identical except for their colors) and arrange them in a row. It is required that no two green balls are adjacent. Find the number of arrangements that meet the requirements.
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Answer: 28357376
Consider a graph with 199 vertices, where each pair of vertices are connected by an edge. 199 of the edges are two-way and the rest of the edges are one-way. We called a quadruple of vertices \(A, B, C, D\) "mutually-connected" if one can move from any vertex to any other vertex among the four via the edges. Find the maximum possible number of such "mutually-connected" quadruples.
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Answer: 33124147