question stringlengths 32 2.14k | final_answer stringlengths 1 3.81k | source_dataset stringclasses 2
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An album contains 25 photos, each showing three people, subject to the following conditions: (1) In each photo, the person on the left is a full brother of the person in the middle, and the person on the right is the daughter of the person in the middle. (2) The person in the middle is different in each of the 25 photo... | \boxed{34} | ycchen/Crystal-Math-Preview | amo-00027 | AMO-Bench | MIT | null | Combinatorics | ["amo/answer_type:number"] |
Let \( a, b, c, d \) be real numbers such that \[4a^4 + \frac{c^4}{16} + b^4 + b^3d^2 + 27 + \frac{d^2}{4} + a^2c^2 = 3c^2 + 24a^2 + 8b^2 + 4d^2.\] Find the minimum value of \[\frac{4a^4 + \frac{c^4}{16} + a^2c^2 + 12a^2 + \frac{3}{2}c^2 + b^4 + \frac{d^4}{4} + 10 + b^2d^2 + 2b^2 + d^2 }{\sqrt{2a^2b^2 + a^2d^2 + 2a^2 +... | \boxed{\frac{9\sqrt{3}}{2}} | ycchen/Crystal-Math-Preview | amo-00013 | AMO-Bench | MIT | null | Analysis | ["amo/answer_type:number"] |
Find $C_{\max}$ such that $\forall x, y, z \in \mathbb{R}$,$$f = 1 + |x+y+z| + |xy+yz+zx| + |xyz| \geq C(|x| + |y| + |z|)$$ | \boxed{\frac{3(16+(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}-1)^2)}{32(\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}-1)}} | ycchen/Crystal-Math-Preview | amo-00028 | AMO-Bench | MIT | null | Algebra | ["amo/answer_type:number"] |
Let $\{a_k\}$ be a sequence of non-negative integers, and $\{b_k\}$ be a sequence of positive real numbers. Given that:
1.$$\left(\sum_{n=1}^{2023} a_n b_n\right)\left( \sum_{k=0}^{673} \frac{1}{(k+675)(1348-k)} \right)= \sum_{k=1}^{674}\frac{1}{k(2k-1)}$$
2.$$\frac{\sum_{k=1}^{1364176}\frac{1}{k(2k-1)}}{\sum_{k=0}^{13... | \boxed{\frac{33497570861567}{2}} | ycchen/Crystal-Math-Preview | amo-00011 | AMO-Bench | MIT | null | Algebra | ["amo/answer_type:number"] |
Let $n$ be a positive integer such that $ 2 \leq n \leq 5 $. For each such $n$, let $A_nB_nC_nD_n$ be a rectangle with vertex $A_n$ at $(-n, 1 + 6 \sqrt{2} - \sqrt{2}n)$, and vertex $B_n$ the reflection of $A_n$ through the origin. The line segments $A_nC_n$ and $B_nD_n$ are parallel to the $x$-axis, and the line segme... | \boxed{68} | ycchen/Crystal-Math-Preview | amo-00014 | AMO-Bench | MIT | null | Geometry | ["amo/answer_type:number"] |
For each positive integer \(k \ge 2\), define \(f(k)\) to be the product of the distinct prime factors of \(k\). All the coefficients of the polynomial function \(9P(x)\) are rational numbers. Suppose \(45 \le P(2025) \le 81\), and there are infinitely many positive integers \(n\) for which \(P(n) = f(n)\).
How many po... | \boxed{42} | ycchen/Crystal-Math-Preview | amo-00022 | AMO-Bench | MIT | null | Number Theory | ["amo/answer_type:number"] |
Let complex numbers \(z_1, z_2, \ldots, z_{2025}\) satisfy
\[
\sum_{k=1}^{2025} z_k = 0,\qquad \sum_{k=1}^{2025} |z_k|^2 = 1.
\]
Find the minimum possible value of
\[
\max_{1 \le i < j \le 2025} |z_i - z_j|.
\] | \boxed{\frac{\sqrt{3}}{45}} | ycchen/Crystal-Math-Preview | amo-00016 | AMO-Bench | MIT | null | Geometry | ["amo/answer_type:number"] |
Let $x_1,x_2,\ldots,x_{2025}$ be real numbers in the interval $[0,1]$, and let $f(x)$ be a real-valued function defined on $[0,1]$. Find the minimum possible value of
\[
\max_{x_1,x_2,\ldots,x_{2025}\in[0,1]}
\left|
\sum_{k=1}^{2025} f(x_k)
-
\left(\sum_{k=1}^{2025} x_k\right)^2
\right|.
\] | \boxed{512578} | ycchen/Crystal-Math-Preview | amo-00002 | AMO-Bench | MIT | null | Analysis | ["amo/answer_type:number"] |
Let \(x,y,z\) be positive real numbers such that \({xy} + {xz} + {yz} \neq 1\), \(\frac{\left( {{x}^{2} - 1}\right) \left( {{y}^{2} - 1}\right) }{xy} + \frac{\left( {{x}^{2} - 1}\right) \left( {{z}^{2} - 1}\right) }{xz} + \frac{\left( {{z}^{2} - 1}\right) \left( {{y}^{2} - 1}\right) }{zy} = 4\). Find the minimum value ... | \boxed{-9126} | ycchen/Crystal-Math-Preview | amo-00008 | AMO-Bench | MIT | null | Algebra | ["amo/answer_type:number"] |
Let \((a_i, b_i)\), for \(1 \leq i \leq 2025\), be 2025 distinct points on the curve
\[3x^2 + 3y^2 = 8 + 4xy + 2(x + y)|x - y|\]
Let \(d_i\) be the distance from the point \((a_i, b_i)\) to the line \(x + y = 6\), and let \(\left\lfloor d_i \right\rfloor\) denote the greatest integer not exceeding \(d_i\).
It is given... | \boxed{2295} | ycchen/Crystal-Math-Preview | amo-00025 | AMO-Bench | MIT | null | Combinatorics | ["amo/answer_type:number"] |
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... | 297 | ycchen/Crystal-Math-Preview | beyondaime-00035 | BeyondAIME | CC0 1.0 | null | Number Theory | null |
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 trapezo... | 24931 | ycchen/Crystal-Math-Preview | beyondaime-00074 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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}+\... | 794430 | ycchen/Crystal-Math-Preview | beyondaime-00083 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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. | 224 | ycchen/Crystal-Math-Preview | beyondaime-00030 | BeyondAIME | CC0 1.0 | null | Algebra | null |
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. | 367236 | ycchen/Crystal-Math-Preview | beyondaime-00081 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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... | 12 | ycchen/Crystal-Math-Preview | beyondaime-00004 | BeyondAIME | CC0 1.0 | null | Number Theory | null |
\(x_1,x_2,\dots,x_{2025}\) 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 \(F(x)\). | 2304 | ycchen/Crystal-Math-Preview | beyondaime-00061 | BeyondAIME | CC0 1.0 | null | Number Theory | null |
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 fac... | 64 | ycchen/Crystal-Math-Preview | beyondaime-00020 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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 \(S\). | 3571 | ycchen/Crystal-Math-Preview | beyondaime-00062 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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. | 576 | ycchen/Crystal-Math-Preview | beyondaime-00046 | BeyondAIME | CC0 1.0 | null | Algebra | null |
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 ... | 2085451 | ycchen/Crystal-Math-Preview | beyondaime-00091 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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,... | 108 | ycchen/Crystal-Math-Preview | beyondaime-00026 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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 observe... | 8700 | ycchen/Crystal-Math-Preview | beyondaime-00066 | BeyondAIME | CC0 1.0 | null | Geometry | null |
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 num... | 1403 | ycchen/Crystal-Math-Preview | beyondaime-00058 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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 \(m \mod 848\). | 46 | ycchen/Crystal-Math-Preview | beyondaime-00017 | BeyondAIME | CC0 1.0 | null | Algebra | null |
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 \(m\). | 22 | ycchen/Crystal-Math-Preview | beyondaime-00009 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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... | 3 | ycchen/Crystal-Math-Preview | beyondaime-00000 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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 \(r + s\). | 6068 | ycchen/Crystal-Math-Preview | beyondaime-00064 | BeyondAIME | CC0 1.0 | null | Algebra | null |
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 consider... | 24 | ycchen/Crystal-Math-Preview | beyondaime-00011 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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 ... | 999499 | ycchen/Crystal-Math-Preview | beyondaime-00084 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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... | 15744 | ycchen/Crystal-Math-Preview | beyondaime-00073 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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 m... | 33124147 | ycchen/Crystal-Math-Preview | beyondaime-00099 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
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 \(G\). | 10320 | ycchen/Crystal-Math-Preview | beyondaime-00070 | BeyondAIME | CC0 1.0 | null | Combinatorics | null |
Given that all vertices of a convex hexagon \(P\) lie on the sides of a unit square, and all its interior angles are equal. Find the maximum possible value of the shortest side length of \(P\). The original answer is in the simplest form of \(a-\frac{\sqrt{m}}{n}\), please give the value of \(a+m+n\). | 7 | ycchen/Crystal-Math-Preview | dapo-13927 | DAPO-17K | Apache 2.0 | null | Geometry | null |
A circle of radius \(2\) is cut into four congruent arcs. The four arcs are joined to form the star figure shown. Find the ratio of the area of the star figure to the area of the original circle. The original answer is in the form \(\frac{k-\pi}{m\pi}\). Please provide the value of \(k + m\). | 5 | ycchen/Crystal-Math-Preview | dapo-14601 | DAPO-17K | Apache 2.0 | null | Geometry | null |
We have 1985 sets. Each of the sets has 45 elements, the union of any two sets has exactly 89 elements. How many elements has the union of all these 1985 sets? Please provide the final integer value of the total number of elements in the union. | 87341 | ycchen/Crystal-Math-Preview | dapo-13835 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. Find the probability that she gets two heads in a row but she sees a second tail before she sees a second head.... | 25 | ycchen/Crystal-Math-Preview | dapo-15099 | DAPO-17K | Apache 2.0 | null | Probability | null |
In trapezoid \(ABCD\), \(BC // AD\), \(AB = AD\), \(\angle ABC = \frac{2\pi}{3}\), \(\angle BCD = \frac{\pi}{2}\). Triangle \(ABD\) is folded along \(BD\), and the projection of point \(A\) onto plane \(BCD\) is point \(P\). Given that the cosine of the angle between \(AB\) and \(CD\) is \(\frac{\sqrt{3}}{6}\), find th... | 3 | ycchen/Crystal-Math-Preview | dapo-12723 | DAPO-17K | Apache 2.0 | null | Geometry | null |
For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\} = x - \lfloor x \rfloor$ to be the fractional part of $x$. For example, $\{3\} = 0$ and $\{4.56\} = 0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x))... | 10 | ycchen/Crystal-Math-Preview | dapo-6135 | DAPO-17K | Apache 2.0 | null | Algebra | null |
Find all $r>0$ such that whenever $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a differentiable function such that $|\operatorname{grad} f(0,0)|=1$ and $|\operatorname{grad} f(u)-\operatorname{grad} f(v)| \leq|u-v|$ for all $u, v \in \mathbb{R}^{2}$, then the maximum of $f$ on the disk $\left\{u \in \mathbb{R}^{2}:|u|... | 3 | ycchen/Crystal-Math-Preview | dapo-13091 | DAPO-17K | Apache 2.0 | null | Analysis | null |
For $x \in R$, $f(x)$ satisfies $f(x)+f(1-x)=1$, $f(x)=2 f\left(\frac{x}{5}\right)$, and for $0 \leq x_{1} \leq x_{2} \leq 1$, we always have $f(x_{1}) \leq f(x_{2})$. Find $f\left(\frac{1}{2022}\right)$. The answer should be in the form $\frac{m}{n}$, where $m$ and $n$ are coprime. Find the value of $m+n$. | 33 | ycchen/Crystal-Math-Preview | dapo-12782 | DAPO-17K | Apache 2.0 | null | Algebra | null |
Two types of pieces, bishops and rooks, are to be placed on a $10 \times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to attack a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that d... | 50 | ycchen/Crystal-Math-Preview | dapo-3455 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Find the smallest positive integer \(t\) with the following property: In a \(100\times 100\) grid, each small square is colored with one color. If the number of squares of each color is at most \(104\), then there exists a \(1\times t\) or \(t\times 1\) rectangle in which the \(t\) small squares contain at least three ... | 12 | ycchen/Crystal-Math-Preview | dapo-11866 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$ | 5 | ycchen/Crystal-Math-Preview | dapo-5590 | DAPO-17K | Apache 2.0 | null | Geometry | null |
The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, find the ratio \(\frac{XQ}{QY}\). The original answer is in the form \(\frac{k}{m}\), please provide the value of \(k + m\). | 5 | ycchen/Crystal-Math-Preview | dapo-14486 | DAPO-17K | Apache 2.0 | null | Geometry | null |
There are 10 distinct positive real numbers. Consider their pairwise sums and pairwise products (45 of each). Given that among these sums, five are equal, find the largest positive integer k such that among these products, k are equal. | 4 | ycchen/Crystal-Math-Preview | dapo-12089 | DAPO-17K | Apache 2.0 | null | Algebra | null |
A man named Juan has three rectangular solids, each having volume $128$. Two of the faces of one solid have areas $4$ and $32$. Two faces of another solid have areas $64$ and $16$. Finally, two faces of the last solid have areas $8$ and $32$. What is the minimum possible exposed surface area of the tallest tower Juan ... | 688 | ycchen/Crystal-Math-Preview | dapo-7025 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Given that $S=\{(i, j) \mid i, j=1,2, \cdots, 100\}$ is the set of $100 \times 100$ integer points on the coordinate plane. Each point in $S$ is colored with one of four given colors. Find the maximum possible number of rectangles with sides parallel to the coordinate axes and with vertices of four distinct colors from... | 9375000 | ycchen/Crystal-Math-Preview | dapo-11007 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
A road company is trying to build a system of highways in a country with $21$ cities. Each highway runs between two cities. A trip is a sequence of distinct cities $C_1,\dots, C_n$, for which there is a highway between $C_i$ and $C_{i+1}$. The company wants to fulfill the following two constraints:
(1) for any ordered... | 192 | ycchen/Crystal-Math-Preview | dapo-7439 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is in the form \(\frac{k}{m}\). P... | 11 | ycchen/Crystal-Math-Preview | dapo-14429 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Given a $2008 \times 2008$ chessboard, where each small square has a distinct color. Fill each small square of the chessboard with one of the four letters $C, G, M, O$. If every $2 \times 2$ subboard contains all four letters $C, G, M, O$, then the chessboard is called a "harmonious chessboard". How many different harm... | 1998 | ycchen/Crystal-Math-Preview | dapo-13905 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Let $N$ be the number of convex $27$-gons up to rotation there are such that each side has length $ 1$ and each angle is a multiple of $2\pi/81$. Find the remainder when $N$ is divided by $23$. | 12 | ycchen/Crystal-Math-Preview | dapo-7639 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
In a 100×100 grid, 300 cells are colored blue, with each row and each column having exactly 3 blue cells. Find the largest positive integer k such that it is always possible to recolor k of the blue cells to red, without forming any red 2×2 square. | 250 | ycchen/Crystal-Math-Preview | dapo-11847 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
There is a \(2024\times2024\) grid. Initially, all cells are white. In one operation, you may choose a row or column in which every cell is white, and color exactly 1000 cells red in that row or column. Find the maximum possible number of red cells after a finite number of operations. | 3048000 | ycchen/Crystal-Math-Preview | dapo-12137 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
A rectangle with perimeter $176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles? | 80 | ycchen/Crystal-Math-Preview | dapo-14223 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Let $a_{1}, a_{2} \ldots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_{1} \ldots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? Please provide th... | 9 | ycchen/Crystal-Math-Preview | dapo-13088 | DAPO-17K | Apache 2.0 | null | Algebra | null |
Caltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be formed with at least two Caltech students as vertices? | 808500 | ycchen/Crystal-Math-Preview | dapo-1452 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
In rectangle $ABCD$, $AB=2$, $AD=1$. Moving point $P$ on side $DC$ (including points $D$ and $C$) and moving point $Q$ on the extension of $CB$ (including point $B$) satisfy $|\overrightarrow{DP}|=|\overrightarrow{BQ}|$. Find the minimum value of the scalar product $\overrightarrow{PA} \cdot \overrightarrow{PQ}$.
The ... | 7 | ycchen/Crystal-Math-Preview | dapo-12461 | DAPO-17K | Apache 2.0 | null | Geometry | null |
From the set of integers $\{1,2,3,\dots,2009\}$, choose $k$ pairs $\{a_i,b_i\}$ with $a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $a_i+b_i$ are distinct and less than or equal to $2009$. Find the maximum possible value of $k$. | 803 | ycchen/Crystal-Math-Preview | dapo-5909 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
In the given diagram, semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, such that they are mutually tangent. If $\overline{CD} \bot \overline{AB}$, find the ratio of the shaded area to the area of a circle with $\overline{CD}$ as its radius. The original answer is in the f... | 5 | ycchen/Crystal-Math-Preview | dapo-16576 | DAPO-17K | Apache 2.0 | null | Geometry | null |
In triangle $A B C$, $A B=1$, $A C=2$, and $B-C=\frac{2 \pi}{3}$. Find the area of triangle $A B C$. The answer should be in the form $\frac{a \sqrt{b}}{c}$. Find the value of $a+b+c$. | 20 | ycchen/Crystal-Math-Preview | dapo-13135 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Suppose $P(x)$ is a monic polynomial of degree $2023$ such that $P(k) = k^{2023}P(1-\frac{1}{k})$ for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n < 2027$ such that $bn-a$ is divisible by the prime $2027$. | 406 | ycchen/Crystal-Math-Preview | dapo-7318 | DAPO-17K | Apache 2.0 | null | Number Theory | null |
The figure below shows a $9 \times 7$ arrangement of $2 \times 2$ squares. Alternate squares of the grid are split into two triangles, with one of the triangles shaded. Find the area of the shaded region. \[ \text{[asy]} \] \[ \text{size}(5\text{cm}); \] \[ \text{defaultpen}(\text{linewidth}(.6)); \] \[ \text{fill}((0... | 64 | ycchen/Crystal-Math-Preview | dapo-5042 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Let $ABC$ be a triangle with sides 3, 4, and 5, and $DEFG$ be a 6-by-7 rectangle. A segment is drawn to divide triangle $ABC$ into a triangle $U_1$ and a trapezoid $V_1$ and another segment is drawn to divide rectangle $DEFG$ into a triangle $U_2$ and a trapezoid $V_2$ such that $U_1$ is similar to $U_2$ and $V_1$ is s... | 35 | ycchen/Crystal-Math-Preview | dapo-5770 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Let the function $f:\bN_+\rightarrow\bN_+$ satisfy that for any positive integers $s,t$, we have $f(t^2f(s))=sf(t)^2$. Find the minimum possible value of $f(1998)$. | 120 | ycchen/Crystal-Math-Preview | dapo-11959 | DAPO-17K | Apache 2.0 | null | Algebra | null |
What is the greatest number of balls with a radius of $\frac{1}{2}$ that can be placed within a rectangular box of size $10 \times 10 \times 1$? | 106 | ycchen/Crystal-Math-Preview | dapo-3641 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
A Parliament of $2000 MP$s decided to ratify the state budget of 200 expenditure items. Each $MP$ prepared a draft budget with what (s)he thinks the maximum possible allocation for each item so that the total expenditure does not exceed a given ceiling, S. For each item, the Parliament approves the maximum expenditure ... | 1991 | ycchen/Crystal-Math-Preview | dapo-10973 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Find the last three digits of
\[2008^{2007^{\cdot^{\cdot^{\cdot ^{2^1}}}}}.\] | 8 | ycchen/Crystal-Math-Preview | dapo-920 | DAPO-17K | Apache 2.0 | null | Number Theory | null |
Several pairwise disjoint isosceles right triangles with legs of length 1 are placed on a 100×100 grid. It is known that the hypotenuse of each right triangle is a diagonal of some unit square, and each side of a unit square is a leg of exactly one right triangle. A unit square whose two diagonals are not the hypotenus... | 2450 | ycchen/Crystal-Math-Preview | dapo-11987 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
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 poin... | 12 | ycchen/Crystal-Math-Preview | dapo-14145 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Two squares of different sizes overlap as shown in the given figure. What is the difference between the non-overlapping areas? | 20 | ycchen/Crystal-Math-Preview | dapo-11815 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Dreamland has 2016 cities. Starcourse Airlines wants to open one-way routes between some pairs of cities so that each city has exactly one outgoing flight. Find the smallest positive integer k such that no matter how the airline opens the routes, it is always possible to divide the 2016 cities into k groups satisfying ... | 57 | ycchen/Crystal-Math-Preview | dapo-12198 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Using six thin iron rods of equal length, a regular tetrahedral frame is welded together, neglecting the thickness of the rods and welding errors. Let the radius of the largest sphere that can be contained within this frame be \(R_{1}\), and the radius of the smallest sphere that can enclose this frame be \(R_{2}\). Fi... | 6 | ycchen/Crystal-Math-Preview | dapo-13716 | DAPO-17K | Apache 2.0 | null | Geometry | null |
A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\tfrac{m}{n},$ where $m$... | 173 | ycchen/Crystal-Math-Preview | dapo-6119 | DAPO-17K | Apache 2.0 | null | Geometry | null |
There are 101 distinct rays in the plane. Any two rays are not collinear, have no common endpoints, and no three are concurrent. Find the maximum number of unordered pairs of rays (a, b) such that a and b intersect, and if their intersection point is C, then the distances from C to the endpoints of a and b are equal. | 2550 | ycchen/Crystal-Math-Preview | dapo-11849 | DAPO-17K | Apache 2.0 | null | Geometry | null |
On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling ... | 375 | ycchen/Crystal-Math-Preview | dapo-5883 | DAPO-17K | Apache 2.0 | null | Algebra | null |
A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ strip of double thickness.... | 593 | ycchen/Crystal-Math-Preview | dapo-5782 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Let positive integers \(a_1,a_2,\cdots,a_{31},b_1,b_2,\cdots,b_{31}\) satisfy: (1) \(a_1<a_2<\cdots<a_{31}\leq2015\), \(b_1<b_2<\cdots<b_{31}\leq 2015\); (2) \(a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}\). Find the maximum value of \(S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|\). | 30720 | ycchen/Crystal-Math-Preview | dapo-12203 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
In February, as winter gives way to spring, a few green buds begin to sprout faintly on the ginkgo tree branches. Delighted by the sight, Dad cooks a large circular spring pancake with a radius of 1 to share with the chicken, dog, crab, and sloth, so as not to waste the beautiful spring scenery. The pancake’s aroma is ... | 30 | ycchen/Crystal-Math-Preview | dapo-12196 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Find the smallest positive integer \(n\) such that there exist \(n\) distinct positive integers \(s_1, s_2, \cdots, s_n\) satisfying
\[
\left(1-\frac{1}{s_1}\right)\left(1-\frac{1}{s_2}\right)\cdots\left(1-\frac{1}{s_n}\right)=\frac{51}{2010}.
\] | 39 | ycchen/Crystal-Math-Preview | dapo-12076 | DAPO-17K | Apache 2.0 | null | Number Theory | null |
In a grid of size 99×101, some squares are colored black and the rest are colored white. If each black square is adjacent to at most one other black square (sharing a common side), what is the maximum number of squares that can be colored black? | 5017 | ycchen/Crystal-Math-Preview | dapo-11982 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on... | 810 | ycchen/Crystal-Math-Preview | dapo-9345 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Suppose there are $n(n \geqslant 5)$ distinct points arranged arbitrarily on a circle, labeled $1,2, \cdots, n$, and denote this arrangement as $S$. For an arrangement, a "descent chain" refers to several consecutive adjacent points on the circle whose labels form a strictly decreasing sequence in clockwise direction (... | 97 | ycchen/Crystal-Math-Preview | dapo-12712 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle{ABC}$? Provide your answer in the form $a\sqrt{b}$, where $a$ and $b$ are integers. Please find the value of a + b. | 6 | ycchen/Crystal-Math-Preview | dapo-14781 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Let the polynomial \( f(x)=x^{2024}+\ds{i=0}{2023}c_ix^i \), where \( c_i\in\{-1,0,1\} \). Denote by \( N \) the number of positive integer roots of \( f(x) \) (including multiplicities). If \( f(x) \) has no negative integer roots, the maximum value of \( N \) is __________. | 10 | ycchen/Crystal-Math-Preview | dapo-12101 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Find the smallest integer \(n \geq 3\) such that for any set of \(n\) points in the plane, no three of which are collinear, there always exist three points that are the vertices of a non-isosceles triangle. | 7 | ycchen/Crystal-Math-Preview | dapo-11948 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Develop necessary and sufficient conditions which ensure that $r_{1},r_{2},r_{3}$ and $r_{1}{}^{2},r_{2}{}^{2}, r_{3}{}^{2}$ are simultaneously roots of the equation $x^{3}+ax^{2}+bx+c=0$.How many possible situations are there in total? | 8 | ycchen/Crystal-Math-Preview | dapo-13271 | DAPO-17K | Apache 2.0 | null | Algebra | null |
\( n \) teams compete in a home-and-away double round-robin tournament (i.e., every two teams play two matches, one of which is a home match for each team). It is known that each team can play multiple away matches in one week (the seven days from Sunday to Saturday), but if a team has a home match in a certain week, t... | 6 | ycchen/Crystal-Math-Preview | dapo-12192 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Let S be a set of 35 elements, and let \(\mathcal{F}\) be a set consisting of some maps from S to itself. For a positive integer k, we say that \(\mathcal{F}\) has property P(k) if for any \(x, y \in S\), there exist k maps \(f_1, f_2, \cdots, f_k\) in \(\mathcal{F}\) (which can be the same) such that \(f_k(\cdots(f_2(... | 595 | ycchen/Crystal-Math-Preview | dapo-12265 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
If a quintuple of integers can have its elements labeled in some order as a, b, c, d, e such that a-b+c-d+e=29, then we call this quintuple "arrangeable". Find all 2017-tuples of integers (n_1,n_2,\cdots,n_{2017}) such that when these 2017 numbers are arranged clockwise on a circle (with n_{2017} adjacent to n_1), any ... | 29 | ycchen/Crystal-Math-Preview | dapo-5394 | DAPO-17K | Apache 2.0 | null | Algebra | null |
Call a polynomial $P\left(x_{1}, \ldots, x_{k}\right)$ good if there exist $2 \times 2$ real matrices $A_{1}, \ldots, A_{k}$ such that $$ P\left(x_{1}, \ldots, x_{k}\right)=\operatorname{det}\left(\sum_{i=1}^{k} x_{i} A_{i}\right) $$ Find all values of $k$ for which all homogeneous polynomials with $k$ variables of deg... | 3 | ycchen/Crystal-Math-Preview | dapo-13093 | DAPO-17K | Apache 2.0 | null | Algebra | null |
There are 100 scenic spots in a city. Each spot earns 1 point in the Tourism Bureau for every tourist it receives. During a 100-day tourist season, 100 tourists visit these spots. Each day, some tourists go out to visit distinct spots (each tourist decides to visit one spot or rest, and no two tourists visit the same s... | 1000 | ycchen/Crystal-Math-Preview | dapo-11716 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
In the Cartesian coordinate plane, consider the hyperbola $\Gamma: \frac{x^{2}}{3}-y^{2}=1$. For any point $P$ in the plane not on $\Gamma$, let $\Omega_{P}$ be the set of all lines through $P$ that intersect $\Gamma$ at two points. For any line $l \in \Omega_{P}$, denote by $M$ and $N$ the two intersection points of $... | 4 | ycchen/Crystal-Math-Preview | dapo-11317 | DAPO-17K | Apache 2.0 | null | Geometry | null |
In the diagram, what is the value of $y$?
[asy]
draw((0,0)--(18,0),black+linewidth(1));
draw((18,0)--(18,-6),black+linewidth(1));
draw((0,0)--(4,6)--(18,-6),black+linewidth(1));
draw((18,0)--(18,-0.5)--(17.5,-0.5)--(17.5,0)--cycle,black+linewidth(1));
label("$80^{\circ}$",(4.5,5),S);
label("$60^{\circ}$",(1,0),NE);
la... | 50 | ycchen/Crystal-Math-Preview | dapo-10675 | DAPO-17K | Apache 2.0 | null | Geometry | null |
Divide a regular 2017-gon into 2015 triangles using 2014 of its diagonals that do not intersect in the interior. Find the maximum possible number of isosceles triangles among these triangles. | 2010 | ycchen/Crystal-Math-Preview | dapo-12225 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is $5$, that $BC=6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine of the central angle o... | 175 | ycchen/Crystal-Math-Preview | dapo-10153 | DAPO-17K | Apache 2.0 | null | Geometry | null |
There is an antelope piece on a chessboard. It can jump from square \((x_1,y_1)\) to square \((x_2,y_2)\) if and only if \(\{|x_1-x_2|,|y_1-y_2|\}=\{3,4\}\). Arrange the numbers from \(1\) to \(10^{12}\) arbitrarily in a \(10^6 \times 10^6\) grid, each number placed exactly in one square. Let \(D\) be the set of number... | 8 | ycchen/Crystal-Math-Preview | dapo-11990 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ such that the sum of the three interior angles at $A$ is $360^\circ$. Thus, the three polygons form a new polygon with $A$ as an interior point. What is the largest possible ... | 21 | ycchen/Crystal-Math-Preview | dapo-16672 | DAPO-17K | Apache 2.0 | null | Geometry | null |
What is the second smallest four-digit number in Pascal's triangle? | 1001 | ycchen/Crystal-Math-Preview | dapo-9406 | DAPO-17K | Apache 2.0 | null | Combinatorics | null |
$A_1A_2A_3A_4$ is a cyclic quadrilateral inscribed in circle $\Omega$, with side lengths $A_1A_2 = 28$, $A_2A_3 =12\sqrt3$, $A_3A_4 = 28\sqrt3$, and $A_4A_1 = 8$. Let $X$ be the intersection of $A_1A_3, A_2A_4$. Now, for $i = 1, 2, 3, 4$, let $\omega_i$ be the circle tangent to segments$ A_iX$, $A_{i+1}X$, and $\Omega$... | 784 | ycchen/Crystal-Math-Preview | dapo-7631 | DAPO-17K | Apache 2.0 | null | Geometry | null |
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