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

2023 Auckland Mathematical Olympiad, 4
Tags: number theory
Which digit must be substituted instead of the star so that the following large number $$\underbrace{66...66}_{2023} \star \underbrace{55...55}_{2023}$$ is divisible by $7$?

1994 Tuymaada Olympiad, 4
Tags: geometry , convex , polyhedron , sphere
Let a convex polyhedron be given with volume $V$ and full surface $S$. Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.

1978 IMO Longlists, 29
Tags: function , algebra
Given a nonconstant function $f : \mathbb{R}^+ \longrightarrow\mathbb{R}$ such that $f(xy) = f(x)f(y)$ for any $x, y > 0$, find functions $c, s : \mathbb{R}^+ \longrightarrow \mathbb{R}$ that satisfy $c\left(\frac{x}{y}\right) = c(x)c(y)-s(x)s(y)$ for all $x, y > 0$ and $c(x)+s(x) = f(x)$ for all $x > 0$.

2019 PUMaC Geometry B, 6
Tags: geometry
Let $\Gamma$ be a circle with center $A$, radius $1$ and diameter $BX$. Let $\Omega$ be a circle with center $C$, radius $1$ and diameter $DY $, where $X$ and $Y$ are on the same side of $AC$. $\Gamma$ meets $\Omega$ at two points, one of which is $Z$. The lines tangent to $\Gamma$ and $\Omega$ that pass through $Z$ cut out a sector of the plane containing no part of either circle and with angle $60^\circ$. If $\angle XY C = \angle CAB$ and $\angle XCD = 90^\circ$, then the length of $XY$ can be written in the form $\tfrac{\sqrt a+\sqrt b}{c}$ for integers $a, b, c$ where $\gcd(a, b, c) = 1$. Find $a + b + c$.

2006 Estonia National Olympiad, 1
Tags: algebra , sum
Calculate the sum $$\frac{1}{1+2^{-2006}}+...+ \frac{1}{1+2^{-1}}+ \frac{1}{1+2^{0}}+ \frac{1}{1+2^{1}}+...+ \frac{1}{1+2^{2006}}$$

2005 MOP Homework, 7
Tags: geometry , ratio , number theory , perimeter , trigonometry , modular arithmetic
Let $a$, $b$, and $c$ be pairwise distinct positive integers, which are side lengths of a triangle. There is a line which cuts both the area and the perimeter of the triangle into two equal parts. This line cuts the longest side of the triangle into two parts with ratio $2:1$. Determine $a$, $b$, and $c$ for which the product abc is minimal.

2024 ELMO Shortlist, C1.5
Tags: combinatorics
Let $m, n \ge 2$ be distinct positive integers. In an infinite grid of unit squares, each square is filled with exactly one real number so that [list] [*]In each $m \times m$ square, the sum of the numbers in the $m^2$ cells is equal. [*]In each $n \times n$ square, the sum of the numbers in the $n^2$ cells is equal. [*]There exist two cells in the grid that do not contain the same number. [/list] Let $S$ be the set of numbers that appear in at least one square on the grid. Find, in terms of $m$ and $n$, the least possible value of $|S|$. [i]Kiran Reddy[/i]

2022 Mexico National Olympiad, 4
Tags: combinatorics , grid
Let $n$ be a positive integer. In an $n\times n$ garden, a fountain is to be built with $1\times 1$ platforms covering the entire garden. Ana places all the platforms at a different height. Afterwards, Beto places water sources in some of the platforms. The water in each platform can flow to other platforms sharing a side only if they have a lower height. Beto wins if he fills all platforms with water. Find the least number of water sources that Beto needs to win no matter how Ana places the platforms.

2005 Serbia Team Selection Test, 2
Tags: number theory
$$problem2$$:Determine the number of 100-digit numbers whose all digits are odd, and in which every two consecutive digits differ by 2

2017 Junior Balkan Team Selection Tests - Moldova, Problem 4
Tags: combinatorics
Find the maximum positive integer $k$ such that there exist $k$ positive integers which do not exceed $2017$ and have the property that every number among them cannot be a power of any of the remaining $k-1$ numbers.

1985 Iran MO (2nd round), 5
Tags: combinatorics , probability
In the Archery with an especial gun, the probability of goal is $90 \%.$ If we continue our work until we goal. [b]i)[/b] What is the probability which exactly $3$ balls consumed. [b]ii)[/b] What is the probability which at least $3$ balls consumed.

2018 Korea USCM, 3
Tags: calculus , integration , function
$\Phi$ is a function defined on collection of bounded measurable subsets of $\mathbb{R}$ defined as $$\Phi(S) = \iint_S (1-5x^2 + 4xy-5y^2 ) dx dy$$ Find the maximum value of $\Phi$.

2015 BMT Spring, 7
Tags: algebra , summation , binomial coefficient
Evaluate $\sum_{k=0}^{37}(-1)^k\binom{75}{2k}$.

1982 IMO Shortlist, 13
Tags: geometry , midpoint , triangle , incircle , concurrency
A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

2015 Tuymaada Olympiad, 7
Tags: geometry , circumcircle
In $\triangle ABC$ points $M,O$ are midpoint of $AB$ and circumcenter. It is true, that $OM=R-r$. Bisector of external $\angle A$ intersect $BC$ at $D$ and bisector of external $\angle C$ intersect $AB$ at $E$. Find possible values of $\angle CED$ [i]D. Shiryaev [/i]

MBMT Team Rounds, 2020.13
Tags: geometry
How many ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ has an area of $p$, where $p$ is a prime number less than $100$? [i]Proposed by Joshua Hsieh[/i]

1995 National High School Mathematics League, 4
Tags: ratio , geometry , similar triangles
Color all points on a plane in red or blue. Prove that there exists two similar triangles, their similarity ratio is $1995$, and apexes of both triangles are in the same color.

1992 Austrian-Polish Competition, 5
Tags: geometry , circles , fixed point , fixed , product
Given a circle $k$ with center $M$ and radius $r$, let $AB$ be a fixed diameter of $k$ and let $K$ be a fixed point on the segment $AM$. Denote by $t$ the tangent of $k$ at $A$. For any chord $CD$ through $K$ other than $AB$, denote by $P$ and Q the intersection points of $BC$ and $BD$ with $t$, respectively. Prove that $AP\cdot AQ$ does not depend on $CD$.

2017 Sharygin Geometry Olympiad, 7
Tags: geometry , concurrency
Let $A_1A_2 \dots A_{13}$ and $B_1B_2 \dots B_{13}$ be two regular $13$-gons in the plane such that the points $B_1$ and $A_{13}$ coincide and lie on the segment $A_1B_{13}$, and both polygons lie in the same semiplane with respect to this segment. Prove that the lines $A_1A_9, B_{13}B_8$ and $A_8B_9$ are concurrent.

2024 May Olympiad, 1
Tags: combinatorics
A $4\times 8$ grid is divided into $32$ unit squares. There are square tiles of sizes $1 \times 1$, $2 \times 2$, $3 \times 3$ and $4 \times 4$. The goal is to completely cover the grid using exactly $n$ of these tiles. [list=a] [*]Is it possible to do this if $n = 19$? [*]Is it possible to do this if $n = 14$? [*]Is it possible to do this if $n = 7$? [/list] [b]Note:[/b] The tiles cannot overlap or extend beyond the grid.

2007 IMO Shortlist, 1
Tags: number theory , divisibility
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$. [i]Author: Stephan Wagner, Austria[/i]

2006 CentroAmerican, 5
Tags: combinatorics
The [i]Olimpia[/i] country is formed by $n$ islands. The most populated one is called [i]Panacenter[/i], and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions: a) No pair of islands is joined by more than one bridge. b) Using the bridges we can reach every island from Panacenter. c) If we want to travel from Panacenter to every other island, in such a way that we use each bridge at most once, the number of inhabitants of the islands we visit is strictly decreasing. Determine the number of ways we can build the bridges.

2024 Chile TST Ibero., 4
Tags: inequalities , algebra , tituslemma
Prove that if \( a \), \( b \), and \( c \) are positive real numbers, then the following inequality holds: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

2021 HMNT, 3
Tags: number theory
Suppose $m$ and $n$ are positive integers for which $\bullet$ the sum of the first $m$ multiples of $n$ is $120$, and $\bullet$ the sum of the first $m^3$ multiples of$ n^3$ is $4032000$. Determine the sum of the first $m^2$ multiples of $n^2$

1999 French Mathematical Olympiad, Problem 5
Tags: triangle , geometry
Prove that the points symmetric to the vertices of a triangle with respect to the opposite side are collinear if and only if the distance from the orthocenter to the circumcenter is twice the circumradius.