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: 1766

2012 Bosnia Herzegovina Team Selection Test, 3
Tags: calculus , integration , modular arithmetic , quadratics , number theory , prime numbers , number theory proposed
Prove that for all odd prime numbers $p$ there exist a natural number $m<p$ and integers $x_1, x_2, x_3$ such that: \[mp=x_1^2+x_2^2+x_3^2.\]

2008 Spain Mathematical Olympiad, 1
Tags: number theory proposed , number theory
Let $p$ and $q$ be two different prime numbers. Prove that there are two positive integers, $a$ and $b$, such that the arithmetic mean of the divisors of $n=p^aq^b$ is an integer.

2014 Iran MO (2nd Round), 1
Tags: number theory proposed , number theory
Find all positive integers $(m,n)$ such that \[n^{n^{n}}=m^{m}.\]

2013 IFYM, Sozopol, 2
Tags: modular arithmetic , algebra , polynomial , quadratics , Vieta , number theory proposed , number theory
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$? [i]Proposed by Mahan Malihi[/i]

2023 German National Olympiad, 6
Tags: algebra , number theory proposed , algebra proposed , ceiling function , cubic equation
The equation $x^3-3x^2+1=0$ has three real solutions $x_1<x_2<x_3$. Show that for any positive integer $n$, the number $\left\lceil x_3^n\right\rceil$ is a multiple of $3$.

2007 Greece Junior Math Olympiad, 3
Tags: number theory proposed , number theory
For an integer $n$, denote $A =\sqrt{n^{2}+24}$ and $B =\sqrt{n^{2}-9}$. Find all values of $n$ for which $A-B$ is an integer.

2000 JBMO ShortLists, 3
Tags: floor function , number theory proposed , number theory
Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$

2004 Iran MO (3rd Round), 21
Tags: number theory , greatest common divisor , modular arithmetic , prime numbers , number theory proposed
$ a_1, a_2, \ldots, a_n$ are integers, not all equal. Prove that there exist infinitely many prime numbers $ p$ such that for some $ k$ \[ p\mid a_1^k \plus{} \dots \plus{} a_n^k.\]

2009 District Round (Round II), 1
Tags: modular arithmetic , number theory proposed , number theory
given a 4-digit number $(abcd)_{10}$ such that both$(abcd)_{10}$and$(dcba)_{10}$ are multiples of $7$,having the same remainder modulo $37$.find $a,b,c,d$.

2012 JBMO TST - Macedonia, 1
Tags: number theory , prime numbers , number theory proposed
Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$, where $n$ is a natural number.

2007 Italy TST, 3
Tags: inequalities , geometry , geometric transformation , reflection , number theory , prime numbers , number theory proposed
Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

2014 All-Russian Olympiad, 1
Tags: number theory proposed , number theory
Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$. Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$. [i]N. Agakhanov[/i]

1979 Canada National Olympiad, 3
Tags: least common multiple , number theory proposed , number theory
Let $a$, $b$, $c$, $d$, $e$ be integers such that $1 \le a < b < c < d < e$. Prove that \[\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16},\] where $[m,n]$ denotes the least common multiple of $m$ and $n$ (e.g. $[4,6] = 12$).

1998 Baltic Way, 5
Tags: modular arithmetic , induction , number theory proposed , number theory
Let $a$ be an odd digit and $b$ an even digit. Prove that for every positive integer $n$ there exists a positive integer, divisible by $2^n$, whose decimal representation contains no digits other than $a$ and $b$.

1983 IMO Longlists, 61
Tags: number theory proposed , number theory
Let $a$ and $b$ be integers. Is it possible to find integers $p$ and $q$ such that the integers $p+na$ and $q +nb$ have no common prime factor no matter how the integer $n$ is chosen ?

2014 Tuymaada Olympiad, 1
Tags: modular arithmetic , number theory , greatest common divisor , least common multiple , relatively prime , number theory proposed , Tuymaada
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained? [i](A. Golovanov)[/i]

2002 Romania Team Selection Test, 2
Tags: algebra , polynomial , induction , Diophantine equation , number theory proposed , number theory
The sequence $ (a_n)$ is defined by: $ a_0\equal{}a_1\equal{}1$ and $ a_{n\plus{}1}\equal{}14a_n\minus{}a_{n\minus{}1}$ for all $ n\ge 1$. Prove that $ 2a_n\minus{}1$ is a perfect square for any $ n\ge 0$.

1989 IberoAmerican, 3
Tags: modular arithmetic , Diophantine equation , number theory proposed , number theory
Show that the equation $2x^2-3x=3y^2$ has infinitely many solutions in positive integers.

2006 Germany Team Selection Test, 3
Tags: modular arithmetic , number theory proposed , number theory
Is the following statement true? For each positive integer $n$, we can find eight nonnegative integers $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$ such that $n=\frac{2^a-2^b}{2^c-2^d}\cdot\frac{2^e-2^f}{2^g-2^h}$.

2014 Middle European Mathematical Olympiad, 7
Tags: number theory , least common multiple , number theory proposed
A finite set of positive integers $A$ is called [i]meanly[/i] if for each of its nonempy subsets the arithmetic mean of its elements is also a positive integer. In other words, $A$ is meanly if $\frac{1}{k}(a_1 + \dots + a_k)$ is an integer whenever $k \ge 1$ and $a_1, \dots, a_k \in A$ are distinct. Given a positive integer $n$, determine the least possible sum of the elements of a meanly $n$-element set.

2005 Flanders Junior Olympiad, 3
Tags: geometry , complex numbers , number theory , prime numbers , number theory proposed
Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.

2010 Contests, 1
Tags: number theory proposed , number theory
Let $f(n)=\sum_{k=0}^{2010}n^k$. Show that for any integer $m$ satisfying $2\leqslant m\leqslant 2010$, there exists no natural number $n$ such that $f(n)$ is divisible by $m$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 1)[/i]

2011 ELMO Shortlist, 4
Tags: modular arithmetic , quadratics , number theory proposed , number theory
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and \[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\] [i]Alex Zhu.[/i] [hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]

1992 Brazil National Olympiad, 7
Tags: number theory proposed , number theory
Find all 4-tuples $(a,b,c,n)$ of naturals such that $n^a + n^b = n^c$

2007 China Western Mathematical Olympiad, 2
Tags: number theory proposed , number theory
Find all natural numbers $n$ such that there exist $ x_1,x_2,\ldots,x_n,y\in\mathbb{Z}$ where $x_1,x_2,\ldots,x_n,y\neq 0$ satisfying: \[x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n \equal{} 0\] \[ny^2 \equal{} x_1^2 \plus{} x_2^2 \plus{} \ldots \plus{} x_n^2\]