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

1984 Balkan MO, 3
Tags: Euler , function , LaTeX , modular arithmetic , number theory , totient function , number theory proposed
Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.

2013 India IMO Training Camp, 1
Tags: modular arithmetic , Diophantine equation , number theory proposed , number theory
A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

1987 IMO Longlists, 60
Tags: number theory proposed , number theory
It is given that $x = -2272$, $y = 10^3+10^2c+10b+a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y.$

1994 Iran MO (2nd round), 3
Tags: number theory proposed , number theory
Let $n >3$ be an odd positive integer and $n=\prod_{i=1}^k p_i^{\alpha_i}$ where $p_i$ are primes and $\alpha_i$ are positive integers. We know that \[m=n(1-\frac{1}{p_1})(1-\frac{1}{p_2})(1-\frac{1}{p_3}) \cdots (1-\frac{1}{p_n}).\] Prove that there exists a prime $P$ such that $P|2^m -1$ but $P \nmid n.$

1996 Baltic Way, 7
Tags: number theory proposed , number theory
A sequence of integers $a_1,a_2,\ldots $ is such that $a_1=1,a_2=2$ and for $n\ge 1$, \[a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. \] Prove that $a_n\not= 0$ for all $n$.

2014 ELMO Shortlist, 8
Tags: function , greatest common divisor , number theory , number theory proposed
Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

1982 IMO Longlists, 38
Tags: symmetry , number theory proposed , number theory
Numbers $u_{n,k} \ (1\leq k \leq n)$ are defined as follows \[u_{1,1}=1, \quad u_{n,k}=\binom{n}{k} - \sum_{d \mid n, d \mid k, d>1} u_{n/d, k/d}.\] (the empty sum is defined to be equal to zero). Prove that $n \mid u_{n,k}$ for every natural number $n$ and for every $k \ (1 \leq k \leq n).$

2010 IberoAmerican Olympiad For University Students, 6
Tags: algebra , polynomial , search , number theory proposed , number theory
Prove that, for all integer $a>1$, the prime divisors of $5a^4-5a^2+1$ have the form $20k\pm1,k\in\mathbb{Z}$. [i]Proposed by Géza Kós.[/i]

2009 China Second Round Olympiad, 3
Tags: number theory , greatest common divisor , number theory proposed
Let $k,l$ be two given integers. Prove that there exist infinite many integers $m\ge k$ such that $\gcd\left(\binom{m}{k},l\right)=1$.

2005 Bulgaria National Olympiad, 1
Tags: number theory proposed , number theory
Determine all triples $\left( x,y,z\right)$ of positive integers for which the number \[ \sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}} \] is an integer .

1999 Turkey MO (2nd round), 1
Tags: number theory proposed , number theory
Find the number of ordered quadruples $(x,y,z,w)$ of integers with $0\le x,y,z,w\le 36$ such that ${{x}^{2}}+{{y}^{2}}\equiv {{z}^{3}}+{{w}^{3}}\text{ (mod 37)}$.

2006 Baltic Way, 16
Tags: number theory proposed , number theory , easy , modular arithmetic
Are there $4$ distinct positive integers such that adding the product of any two of them to $2006$ yields a perfect square?

2005 Germany Team Selection Test, 1
Tags: number theory proposed , number theory
[b](a)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ ends with the string $2004$, followed by a number of digits from the set $\left\{0;\;4\right\}$ ? [b](b)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ starts with the string $2004$ ?

1951 Miklós Schweitzer, 8
Tags: number theory , least common multiple , floor function , number theory proposed
Given a positive integer $ n>3$, prove that the least common multiple of the products $ x_1x_2\cdots x_k$ ($ k\geq 1$) whose factors $ x_i$ are positive integers with $ x_1\plus{}x_2\plus{}\cdots\plus{}x_k\le n$, is less than $ n!$.

2007 Indonesia TST, 4
Tags: number theory , greatest common divisor , number theory proposed
Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$.

2009 Indonesia TST, 4
Tags: number theory proposed , number theory
Prove that there exist infinitely many positive integers $ n$ such that $ n!$ is not divisible by $ n^2\plus{}1$.

1999 Korea - Final Round, 3
Tags: modular arithmetic , number theory proposed , number theory
Find all intengers n such that $2^n - 1$ is a multiple of 3 and $(2^n - 1)/3$ is a divisor of $4m^2 + 1$ for some intenger m.

2010 All-Russian Olympiad, 1
Tags: number theory proposed , number theory
If $n \in \mathbb{N} n > 1$ prove that for every $n$ you can find $n$ consecutive natural numbers the product of which is divisible by all primes not exceeding $2n+1$, but is not divisible by any other primes.

2002 Baltic Way, 17
Tags: number theory proposed , number theory
Show that the sequence \[\binom{2002}{2002},\binom{2003}{2002},\binom{2004}{2002},\ldots \] considred modulo $2002$, is periodic.

2013 India National Olympiad, 2
Tags: number theory , greatest common divisor , number theory proposed
Find all $m,n\in\mathbb N$ and primes $p\geq 5$ satisfying \[m(4m^2+m+12)=3(p^n-1).\]

2000 Korea - Final Round, 1
Tags: pigeonhole principle , quadratics , number theory , number theory proposed
Prove that for any prime $p$, there exist integers $x,y,z,$ and $w$ such that $x^2+y^2+z^2-wp=0$ and $0<w<p$

2016 Middle European Mathematical Olympiad, 7
Tags: number theory , number theory proposed , Parity , Digits
A positive integer $n$ is [i]Mozart[/i] if the decimal representation of the sequence $1, 2, \ldots, n$ contains each digit an even number of times. Prove that: 1. All Mozart numbers are even. 2. There are infinitely many Mozart numbers.

2014 CentroAmerican, 1
Tags: number theory , prime numbers , number theory proposed
A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

2012 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: number theory , prime factorization , number theory proposed
An integer $n$ is called [i]apocalyptic[/i] if the addition of $6$ different positive divisors of $n$ gives $3528$. For example, $2012$ is apocalyptic, because it has six divisors, $1$, $2$, $4$, $503$, $1006$ and $2012$, that add up to $3528$. Find the smallest positive apocalyptic number.

2010 Switzerland - Final Round, 7
Tags: number theory proposed , number theory
Let $ m$, $ n$ be natural numbers such that $ m\plus{}n\plus{}1$ is prime and divides $ 2(m^2\plus{}n^2)\minus{}1$. Prove that $ m\equal{}n$.