This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2019 Tournament Of Towns, 7
Tags: grid , coloring , combinatorics
Peter has a wooden square stamp divided into a grid. He coated some $102$ cells of this grid with black ink. After that, he pressed this stamp $100$ times on a list of paper so that each time just those $102$ cells left a black imprint on the paper. Is it possible that after his actions the imprint on the list is a square $101 \times 101$ such that all the cells except one corner cell are black? (Alexsandr Gribalko)

2004 Spain Mathematical Olympiad, Problem 3
Tags: functional equation , algebra
Represent for $\mathbb {Z}$ the set of all integers. Find all of the functions ${f:}$ $ \mathbb{Z} \rightarrow \mathbb{Z}$ such that for any ${x,y}$ integers, they satisfy: ${f(x + f(y)) = f(x) - y.}$

2022 MMATHS, 8
Tags: number theory
In the number puzzle below, each cell contains a digit, each cell in the same bolded region has the same digit, and cells in different bolded regions have different digits. The answers to the clues are to be read as three-, four-, or five-digit numbers. Find the unique solution to the puzzle, given that no answer to any clue has a leading $0$. [img]https://cdn.artofproblemsolving.com/attachments/b/a/23514673819aea46c30fd2947f8c82710a1fb3.png[/img]

2021 Miklós Schweitzer, 5
Tags: real analysis
Let $f(x)=\frac{1+\cos(2 \pi x)}{2}$, for $x \in \mathbb{R}$, and $f^n=\underbrace{ f \circ \cdots \circ f}_{n}$. Is it true that for Lebesgue almost every $x$, $\lim_{n \to \infty} f^n(x)=1$?

2024 Pan-African, 6
Tags: number theory
Find all integers $n$ for which $n^7-41$ is the square of an integer

1974 Poland - Second Round, 3
Tags: 3d geometry , tetrahedron , angle bisector , geometry
Prove that the orthogonal projections of the vertex $ D $ of the tetrahedron $ ABCD $ onto the bisectors of the internal and external dihedral angles at the edges $ \overline{AB} $, $ \overline{BC} $ and $ \overline{CA} $ belong to one plane .

2014 District Olympiad, 3
Tags: geometry
The medians $AD, BE$ and $CF$ of triangle $ABC$ intersect at $G$. Let $P$ be a point lying in the interior of the triangle, not belonging to any of its medians. The line through $P$ parallel to $AD$ intersects the side $BC$ at $A_{1}$. Similarly one defines the points $B_{1}$ and $C_{1}$. Prove that \[ \overline{A_{1}D}+\overline{B_{1}E}+\overline{C_{1}F}=\frac{3}{2}\overline{PG} \]

2014 Spain Mathematical Olympiad, 3
Tags: combinatorics
$60$ points are on the interior of a unit circle (a circle with radius $1$). Show that there exists a point $V$ on the circumference of the circle such that the sum of the distances from $V$ to the $60$ points is less than or equal to $80$.

2023 South East Mathematical Olympiad, 5
Tags: geometry
Let $AB$ be a chord of the semicircle $O$ (not the diameter). $M$ is the midpoint of $AB$, and $D$ is a point lies on line $OM$ ($D$ is outside semicircle $O$). Line $l$ passes through $D$ and is parallel to $AB$. $P, Q$ are two points lie on $l$ and $PO$ meets semicircle $O$ at $C$. If $\angle PCD=\angle DMC$, and $M$ is the orthocentre of $\triangle OPQ$. Prove that the intersection of $AQ$ and $PB$ lies on semicircle $O$.

1992 All Soviet Union Mathematical Olympiad, 565
Tags: table , combinatorics , combinatorial geometry
An $m \times n$ rectangle is divided into mn unit squares by lines parallel to its sides. A gnomon is the figure of three unit squares formed by deleting one unit square from a $2 \times 2$ square. For what $m, n$ can we divide the rectangle into gnomons so that no two gnomons form a rectangle and no vertex is in four gnomons?

2019 Greece Team Selection Test, 3
Tags: number theory
Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

2022 Brazil EGMO TST, 8
Tags: number theory
Find all pairs $(a,b)$ of positive integers, such that for [b]every[/b] $n$ positive integer, the equality $a^n+b^n=c_n^{n+1}$ is true, for some $c_n$ positive integer.

2019 Online Math Open Problems, 7
Tags:
At a concert $10$ singers will perform. For each singer $x$, either there is a singer $y$ such that $x$ wishes to perform right after $y$, or $x$ has no preferences at all. Suppose that there are $n$ ways to order the singers such that no singer has an unsatisfied preference, and let $p$ be the product of all possible nonzero values of $n$. Compute the largest nonnegative integer $k$ such that $2^k$ divides $p$. [i]Proposed by Gopal Goel[/i]

2019 Abels Math Contest (Norwegian MO) Final, 1
Tags: combinatorics , square grid
You have an $n \times n$ grid of empty squares. You place a cross in all the squares, one at a time. When you place a cross in an empty square, you receive $i+j$ points if there were $i$ crosses in the same row and $j$ crosses in the same column before you placed the new cross. Which are the possible total scores you can get?

2024 MMATHS, 1
Tags:
Let $f$ be a function over the domain of all positive real numbers such that $$f(x)=\frac{1-\sqrt{x}}{1+\sqrt{x}}$$ Now, let $g(x)$ be a function given by $$g(x)=f(x)^{\tfrac{2f\left(\tfrac{1}{x}\right)}{f(x)}}$$ $g(100)$ can be expressed as a fraction $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime integers. What is the sum of $a$ and $b$?

1987 Tournament Of Towns, (154) 5
Tags: algebra , inequalities
We are given three non-negative numbers $A , B$ and $C$ about which it is known that $$A^4 + B^4 + C^4 \le 2(A^2B^2 + B^2C^2 + C^2A^2)$$ (a) Prove that each of $A, B$ and $C$ is not greater than the sum of the others. (b) Prove that $A^2 + B^2 + C^2 \le 2(AB + BC + CA)$ . (c) Does the original inequality follow from the one in (b)? (V.A. Senderov , Moscow)

2014 Middle European Mathematical Olympiad, 1
Tags: function , algebra , functional equation , fe , real
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\] holds for all $x,y \in \mathbb{R}$.

2011 QEDMO 9th, 7
Tags: functional equation , functional , algebra
Find all functions $f: R\to R$, such that $f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)$ for all $x, y \in R$.

2018 Tajikistan Team Selection Test, 2
Tags:
Problem 2. Prove that for every n≥3, there exists a convex polygon with n sides, such that one can divide it into n-2 triangles that are all similar, but pairwise non-congruent. [color=#00f]Moved to HSO. ~ oVlad[/color]

1979 All Soviet Union Mathematical Olympiad, 278
Tags: algebra , inequalities
Prove that for the arbitrary numbers $x_1, x_2, ... , x_n$ from the $[0,1]$ segment $$(x_1 + x_2 + ...+ x_n + 1)^2 \ge 4(x_1^2 + x_2^2 + ... + x_n^2)$$

2025 Harvard-MIT Mathematics Tournament, 4
Tags: geometry
A semicircle is inscribed in another semicircle if the smaller semicircle’s diameter is a chord of the larger semicircle, and the smaller semicircle’s arc is tangent to the diameter of the larger semicircle. Semicircle $S_1$ is inscribed in a semicircle $S_2,$ which is inscribed in another semicircle $S_3.$ The radii of $S_1$ and $S_3$ are $1$ and $10,$ respectively, and the diameters of $S_1$ and $S_3$ are parallel. The endpoints of the diameter of of $S_3$ are $A$ and $B,$ and $S_2$'s arc is tangent to $AB$ at $C.$ Compute $AC \cdot CB.$ [center] [img] https://cdn.artofproblemsolving.com/attachments/9/6/ad8c82afe131103793cb2684b45c6d20b00ef0.png [/img] [/center]

2006 Tournament of Towns, 6
Tags: combinatorics , number theory
On a circumference at some points sit $12$ grasshoppers. The points divide the circumference into $12$ arcs. By a signal each grasshopper jumps from its point to the midpoint of its arc (in clockwise direction). In such way new arcs are created. The process repeats for a number of times. Can it happen that at least one of the grasshoppers returns to its initial point after a) $12$ jumps? (4) a) $13$ jumps? (3)

2004 IberoAmerican, 1
Tags: modular arithmetic , algebra , system of equations , number theory
Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a\plus{}b$ and $ 201a\plus{}b$ are both perfect squares.

2004 Junior Balkan MO, 3
Tags: quadratic , modular arithmetic , number theory
If the positive integers $x$ and $y$ are such that $3x + 4y$ and $4x + 3y$ are both perfect squares, prove that both $x$ and $y$ are both divisible with $7$.

2006 Greece JBMO TST, 2
Tags: number theory , prime number , prime , exponential
Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.