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

2017 Turkey Junior National Olympiad, 1

Find all triplets of positive integers $(a,b,c)$ for which the number $3^a+3^b+3^c$ is a perfect square.

2004 Postal Coaching, 18

Let $0 = a_1 < a_2 < a_3 < \cdots < a_n < 1$ and $0 = b_1 < b_2 < b_3 \cdots < b_m < 1$ be real numbers such that for no $a_j$ and $b_k$ the relation $a_j + b_k = 1$ is satisfied. Prove that if the $mn$ numbers ${\ a_j + b_k : 1 \leq j \leq n , 1 \leq k \leq m \}}$ are reduced modulo $1$, then at least $m+n -1$ residues will be distinct.

1990 IMO Longlists, 4

Find the minimal value of the function \[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]

1997 South africa National Olympiad, 6

Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)

2012 District Olympiad, 3

Let $G$ a $n$ elements group. Find all the functions $f:G\rightarrow \mathbb{N}^*$ such that: (a) $f(x)=1$ if and only if $x$ is $G$'s identity; (b) $f(x^k)=\frac{f(x)}{(f(x),k)}$ for any divisor $k$ of $n$, where $(r,s)$ stands for the greatest common divisor of the positive integers $r$ and $s$.

KoMaL A Problems 2021/2022, A. 806

Four distinct lines are given in the plane, which are not concurrent and no three of which are parallel. Prove that it is possible to find four points in the plane, $A,B,C,$ and $D$ with the following properties: [list=1] [*]$A,B,C,$ and $D$ are collinear in this order; [*]$AB=BC=CD$; [*]with an appropriate order of the four given lines, $A$ is on the first, $B$ is on the second, $C$ is on the third and $D$ is on the fourth line. [/list] [i]Proposed by Kada Williams, Cambridge[/i]

2008 Greece Junior Math Olympiad, 3

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Find the greatest value of positive integer $ x$ , such that the number $ A\equal{} 2^{182} \plus{} 4^x \plus{} 8^{700}$ is a perfect square .

2013 Balkan MO Shortlist, C2

Some squares of an $n \times n$ chessboard have been marked ($n \in N^*$). Prove that if the number of marked squares is at least $n\left(\sqrt{n} + \frac12\right)$, then there exists a rectangle whose vertices are centers of marked squares.

2018 AIME Problems, 13

Tags: probability , dice
Misha rolls a standard, fair six-sided die until she rolls $1$-$2$-$3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2025 Romanian Master of Mathematics, 6

Let $k$ and $m$ be integers greater than $1$. Consider $k$ pairwise disjoint sets $S_1,S_2, \cdots S_k$; each of these sets has exactly $m+1$ elements, one of which is red and the other $m$ are all blue. Let $\mathcal{F}$ be the family of all subsets $F$ of $S_1 \bigcup S_2\bigcup \cdots S_k$ such that, for every $i$ , the intersection $F \bigcap S_i$ is monochromatic; the empty set is also monochromatic. Determine the largest cardinality of a subfamily $\mathcal{G} \subseteq \mathcal{F}$, no two sets of which are disjoint. [i]Proposed by Russia, Andrew Kupavskii and Maksim Turevskii[/i]

2023 Thailand TSTST, 4

Find all pairs $(p, n)$ with $n>p$, consisting of a positive integer $n$ and a prime $p$, such that $n^{n-p}$ is an $n$-th power of a positive integer.

2023 Bundeswettbewerb Mathematik, 1

Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.

2003 Iran MO (2nd round), 1

We call the positive integer $n$ a $3-$[i]stratum[/i] number if we can divide the set of its positive divisors into $3$ subsets such that the sum of each subset is equal to the others. $a)$ Find a $3-$stratum number. $b)$ Prove that there are infinitely many $3-$stratum numbers.

2008 Indonesia TST, 3

Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$. (i) Prove that $CK$ is the angle bisector of $\angle ACB$. (ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.

2019 Polish Junior MO Finals, 1.

Let $a$, $b$ be the positive integers greater than $1$. Prove that if $$ \frac{a}{b},\; \frac{a-1}{b-1} $$ differ by 1, then both are integers.

2017 Vietnam Team Selection Test, 1

There are $44$ distinct holes in a line and $2017$ ants. Each ant comes out of a hole and crawls along the line with a constant speed into another hole, then comes in. Let $T$ be the set of moments for which the ant comes in or out of the holes. Given that $|T|\leq 45$ and the speeds of the ants are distinct. Prove that there exists two ants that don't collide.

1978 AMC 12/AHSME, 30

In a tennis tournament, $n$ women and $2n$ men play, and each player plays exactly one match with every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is $7/5$, then $n$ equals $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad \textbf{(E) }\text{none of these}$

2024 Lusophon Mathematical Olympiad, 6

A positive integer $n$ is called $oeirense$ if there exist two positive integers $a$ and $b$, not necessarily distinct, such that $n=a^2+b^2$. Determine the greatest integer $k$ such that there exist infinitely many positive integers $n$ such that $n$, $n+1$, $\dots$, $n+k$ are oeirenses.

2016 Harvard-MIT Mathematics Tournament, 32

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How many equilateral hexagons of side length $\sqrt{13}$ have one vertex at $(0,0)$ and the other five vertices at lattice points? (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but not self-intersecting.)

2011 Today's Calculation Of Integral, 720

Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.

2013 Hitotsubashi University Entrance Examination, 1

Find all pairs $(p,\ q)$ of positive integers such that $3p^3-p^2q-pq^2+3q^3=2013.$

2022 Belarus - Iran Friendly Competition, 4

Tags: geometry
From a point $S$, which lies outside the circle $\Omega$, tangent lines $SA$ and $SB$ to that circle are drawn. On the chord $AB$ an arbitrary point $K$ is chosen. $SK$ intersects $\Omega$ at points $P$ and $Q$, and chords $RT$ and $UW$ pass through $K$ such that $W, Q$ and $T$ lie in the same half-plane with respect to $AB$. Lines $WR$ and $TU$ intersect chord $AB$ at $C$ and $D$, and $M$ is the midpoint of $PQ$. Prove that $\angle AMC = \angle BMD$

2025 Kyiv City MO Round 2, Problem 4

A square \( K = 2025 \times 2025 \) is given. We define a [i]stick[/i] as a rectangle where one of its sides is \( 1 \), and the other side is a positive integer from \( 1 \) to \( 2025 \). Find the largest positive integer \( C \) such that the following condition holds: [list] [*] If several sticks with a total area not exceeding \( C \) are taken, it is always possible to place them inside the square \( K \) so that each stick fully completely covers an integer number of \( 1 \times 1 \) squares, and no \( 1 \times 1 \) square is covered by more than one stick. [/list] [i](Basically, you can rotate sticks, but they have to be aligned by lines of the grid)[/i] [i]Proposed by Anton Trygub[/i]

2020/2021 Tournament of Towns, P7

An integer $n > 2$ is given. Peter wants to draw $n{}$ arcs of length $\alpha{}$ of great circles on a unit sphere so that they do not intersect each other. Prove that [list=a] [*]for all $\alpha<\pi+2\pi/n$ it is possible; [*]for all $\alpha>\pi+2\pi/n$ it is impossible; [/list] [i]Ilya Bogdanov[/i]

LMT Team Rounds 2021+, A12 B18

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There are $23$ balls on a table, all of which are either red or blue, such that the probability that there are $n$ red balls and $23-n$ blue balls on the table ($1 \le n \le 22$) is proportional to $n$. (e.g. the probability that there are $2$ red balls and $21$ blue balls is twice the probability that there are $1$ red ball and $22$ blue balls.) Given that the probability that the red balls and blue balls can be arranged in a line such that there is a blue ball on each end, no two red balls are next to each other, and an equal number of blue balls can be placed between each pair of adjacent red balls is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a+b$. Note: There can be any nonzero number of consecutive blue balls at the ends of the line. [i]Proposed by Ada Tsui[/i]