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

2011 IFYM, Sozopol, 8
Tags: number theory , rational number
Let $a$ and $b$ be some rational numbers and there exist $n$, such that $\sqrt[n]{a}+\sqrt[b]{b}$ is also a rational number. Prove that $\sqrt[n]{a}$ is a rational number.

2023 Azerbaijan BMO TST, 1
Tags: number theory
Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?

1967 German National Olympiad, 5
Tags: number theory , diophantine , diophantine equation
For each natural number $n$, determine the number $A(n)$ of all integer nonnegative solutions the equation $$5x + 2y + z = 10n.$$

1961 Putnam, A1
Tags: analytic geometry , intersection of set
The graph of the equation $x^y =y^x$ in the first quadrant consists of a straight line and a curve. Find the coordinates of the intersection of the line and the curve.

1995 Iran MO (2nd round), 3
Tags: combinatorics
Let $k$ be a positive integer. $12k$ persons have participated in a party and everyone shake hands with $3k+6$ other persons. We know that the number of persons who shake hands with every two persons is a fixed number. Find $k.$

2021 USA IMO Team Selection Test, 3
Tags: algebra , functional equation
Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the inequality \[ f(y) - \left(\frac{z-y}{z-x} f(x) + \frac{y-x}{z-x}f(z)\right) \leq f\left(\frac{x+z}{2}\right) - \frac{f(x)+f(z)}{2} \] for all real numbers $x < y < z$. [i]Proposed by Gabriel Carroll[/i]

1967 IMO Longlists, 7
Tags: algebra , system of equations , polynomial
Find all real solutions of the system of equations: \[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$

2015 SGMO, Q3
Tags: number theory
$a_n,b_n,c_n$ are three sequences of positive integers satisfying $$\prod_{d|n}a_d=2^n-1,\prod_{d|n}b_d=\frac{3^n-1}{2},\prod_{d|n}c_d=\gcd(2^n-1,\frac{3^n-1}{2})$$ for all $n\in \mathbb{N}$. Prove that $\gcd(a_n,b_n)|c_n$ for all $n\in \mathbb{N}$

Ukrainian From Tasks to Tasks - geometry, 2013.9
Tags: geometry , rhombus , trisection
The perpendicular bisectors of the sides $AB$ and $CD$ of the rhombus $ABCD$ are drawn. It turned out that they divided the diagonal $AC$ into three equal parts. Find the altitude of the rhombus if $AB = 1$.

2016 ASDAN Math Tournament, 8
Tags: team test
Let $ABC$ be a triangle with $AB=24$, $BC=30$, and $AC=36$. Point $M$ lies on $BC$ such that $BM=12$, and point $N$ lies on $AC$ such that $CN=20$. Let $X$ be the intersection of $AM$ and $BN$ and let line $CX$ intersect $AB$ at point $L$. Compute $$\frac{AX}{XM}+\frac{BX}{XN}+\frac{CX}{XL}.$$

2001 India IMO Training Camp, 3
Tags: combinatorics
Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$.

2025 India STEMS Category B, 4
Tags: functional equation
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$, \[xf(y+x)+(y+x)f(y)=f(x^2+y^2)+2f(xy)\] [i]Proposed by Aritra Mondal[/i]

1988 India National Olympiad, 8
Tags: vector , geometry
A river flows between two houses $ A$ and $ B$, the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from $ A$ to $ B$, using the bridge to cross the river may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks.

2022 SG Originals, Q3
Tags: functional equation , number theory , factorial , function
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$. [i]Proposed by DVDthe1st[/i]

2024 CMIMC Algebra and Number Theory, 3
Tags: algebra
The positive integer $8833$ has the property that $8833 = 88^2 + 33^2.$ Find the (unique) other four-digit positive integer $\overline{abcd}$ where $\overline{abcd} = (\overline{ab})^2 + (\overline{cd})^2.$ [i]Proposed by Allen Yang[/i]

1984 IMO Longlists, 61
Tags: probability , expected value , combinatorics
A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses.

2007 Purple Comet Problems, 18
Tags: rotation
Let $S$ be the graph of $y=x^3$, and $T$ be the graph of $y=\sqrt[3]{y}$. Let $S^*$ be $S$ rotated around the origin $15$ degrees clockwise, and $T^*$ be T rotated around the origin 45 degrees counterclockwise. $S^*$ and $T^*$ will intersect at a point in the first quadrant a distance $M+\sqrt{N}$ from the origin where $M$ and $N$ are positive integers. Find $M+N$.

2010 India Regional Mathematical Olympiad, 4
Tags:
Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.

2020 Peru IMO TST, 1
Tags: number theory , perfect power , prime number
Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that: For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that: $$m+a_1b+a_2b^2+\cdots+a_nb^n$$ Is a perfect power. Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$

2012 Indonesia TST, 2
Tags: incenter , circumcircle , geometry
Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.

2021 Stanford Mathematics Tournament, 7
Tags: geometry
An $n$-sided regular polygon with side length $1$ is rotated by $\frac{180^o}{n}$ about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a $2n$-sided regular polygon with side length $\frac{1-tan^2 10^o}{2}$. What is the value of $n$?

2021 Alibaba Global Math Competition, 7
Tags: convergence , sobolev space , continuity
A subset $Q \subset H^s(\mathbb{R})$ is said to be equicontinuous if for any $\varepsilon>0$, $\exists \delta>0$ such that \[\|f(x+h)-f(x)\|_{H^s}<\varepsilon, \quad \forall \vert h\vert<\delta, \quad f \in Q.\] Fix $r<s$, given a bounded sequence of functions $f_n \in H^s(\mathbb{R}$. If $f_n$ converges in $H^r(\mathbb{R})$ and equicontinuous in $H^s(\mathbb{R})$, show that it also converges in $H^s(\mathbb{R})$.

2021 VIASM Math Olympiad Test, Problem 3
Tags: combinatorics
Given the positive integer $n$. Let $X = \{1, 2,..., n\}$. For each nonempty subset $A$ of $X$, set $r(A) = max_A - min_A$, where $max_A, min_A$ are the greatest and smallest elements of $A$, respectively. Find the mean value of $r(A)$ when $A$ runs on subsets of $X$.

1982 Poland - Second Round, 5
Tags: number theory , divide , divisible
Let $ q $ be an even positive number. Prove that for every natural number $ n $ number $q^{(q+1)^n}+1$ is divisible by $ (q + 1)^{n+1} $ but not divisible by $ (q + 1)^{n+2} $.

2002 Irish Math Olympiad, 4
Tags: floor function , algebra
Let $ \alpha\equal{}2\plus{}\sqrt{3}$. Prove that $ \alpha^n\minus{}[\alpha^n]\equal{}1\minus{}\alpha^{\minus{}n}$ for all $ n \in \mathbb{N}_0$.