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

2007 Korea Junior Math Olympiad, 8
Tags: number theory , prime , positive integer , divisor , number theory proposed
Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$). Prove that there are infi nite number of [i]Primes of the Year[/i].

2013 Middle European Mathematical Olympiad, 8
Tags: number theory proposed , number theory
The expression \[ \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \] is written on the blackboard. Two players, $ A $ and $ B $, play a game, taking turns. Player $ A $ takes the first turn. In each turn, the player on turn replaces a symbol $ \Box $ by a positive integer. After all the symbols $\Box$ are replace, player $A$ replaces each of the signs $\pm$ by either + or -, independently of each other. Player $ A $ wins if the value of the expression on the blackboard is not divisible by any of the numbers $ 11, 12, \cdots, 18 $. Otherwise, player $ B$ wins. Determine which player has a winning strategy.

2013 JBMO TST - Turkey, 2
Tags: modular arithmetic , number theory , prime numbers , number theory proposed
[b]a)[/b] Find all prime numbers $p, q, r$ satisfying $3 \nmid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares. [b]b)[/b] Do there exist prime numbers $p, q, r$ such that $3 \mid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares?

2000 Austrian-Polish Competition, 4
Tags: number theory proposed , number theory
Find all positive integers $N$ having only prime divisors $2,5$ such that $N+25$ is a perfect square.

2010 Contests, 1
Tags: quadratics , number theory proposed , number theory
Solve in the integers the diophantine equation $$x^4-6x^2+1 = 7 \cdot 2^y.$$

2012 Kosovo Team Selection Test, 5
Tags: number theory , greatest common divisor , number theory proposed
Prove that the equation \[\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\] has infinitly many natural solutions

2014 All-Russian Olympiad, 3
Tags: number theory proposed , number theory
Positive rational numbers $a$ and $b$ are written as decimal fractions and each consists of a minimum period of 30 digits. In the decimal representation of $a-b$, the period is at least $15$. Find the minimum value of $k\in\mathbb{N}$ such that, in the decimal representation of $a+kb$, the length of period is at least $15$. [i]A. Golovanov[/i]

2013 China Team Selection Test, 2
Tags: pigeonhole principle , number theory proposed , number theory
For the positive integer $n$, define $f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|$. Let $\{n_i\}$ be a strictly increasing sequence of positive integers. $C$ is a constant such that $f(n_i)<\dfrac C{n_i^2}$ for all $i\in\{1,2,\ldots\}$. Show that there exists a real number $q>1$ such that $n_i\geqslant q^{i-1}$ for all $i\in\{1,2,\ldots \}$.

2013 ELMO Shortlist, 6
Tags: function , number theory proposed , number theory
Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

2001 Moldova Team Selection Test, 5
Tags: number theory proposed , number theory
Find $ a,b,c \in N$ such that $ ab$ divides $ a^2\plus{}b^2\plus{}1$.

2014 Contests, 4
Tags: number theory proposed , number theory
Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.

2011 All-Russian Olympiad Regional Round, 11.2
Tags: number theory proposed , number theory
2011 non-zero integers are given. It is known that the sum of any one of them with the product of the remaining 2010 numbers is negative. Prove that if all numbers are split arbitrarily into two groups, the sum of the two products will also be negative. (Authors: N. Agahanov & I. Bogdanov)

1982 IMO Longlists, 9
Tags: function , floor function , number theory proposed , number theory
Given any two real numbers $\alpha$ and $\beta , 0 \leq \alpha < \beta \leq 1$, prove that there exists a natural number $m$ such that \[\alpha < \frac{\phi(m)}{m} < \beta.\]

2007 Romania Team Selection Test, 2
Tags: limit , probability , number theory proposed , number theory
The world-renowned Marxist theorist [i]Joric[/i] is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer $n$, he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the [i]defect[/i] of the number $n$. Determine the average value of the defect (over all positive integers), that is, if we denote by $\delta(n)$ the defect of $n$, compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\] [i]Iurie Boreico[/i]

2014 China Team Selection Test, 3
Tags: number theory proposed , number theory , Diophantine equation , China TST
Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

2009 Italy TST, 3
Tags: geometry , number theory proposed , number theory
Find all pairs of integers $(x,y)$ such that \[ y^3=8x^6+2x^3y-y^2.\]

1983 Vietnam National Olympiad, 1
Tags: modular arithmetic , number theory , greatest common divisor , number theory proposed
Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?

1992 Iran MO (2nd round), 3
Tags: function , induction , number theory proposed , number theory
Let $X \neq \varnothing$ be a finite set and let $f: X \to X$ be a function such that for every $x \in X$ and a fixed prime $p$ we have $f^p(x)=x.$ Let $Y=\{x \in X | f(x) \neq x\}.$ Prove that the number of the members of the set $Y$ is divisible by $p.$ [i]Note.[/i] ${f^p(x)=x = \underbrace{f(f(f(\cdots ((f}_{ p \text{ times}}(x) ) \cdots )))} .$

2012 Romanian Master of Mathematics, 4
Tags: modular arithmetic , function , number theory , number theory proposed
Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not. [i](Russia) Valery Senderov[/i]

2002 Federal Competition For Advanced Students, Part 1, 1
Tags: number theory proposed , number theory
Determine all integers $a$ and $b$ such that \[(19a + b)^{18} + (a + b)^{18} + (a + 19b)^{18}\] is a perfect square.

2014 All-Russian Olympiad, 2
Tags: number theory , number theory proposed
Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$ and $b+2$. He then writes all six pairwise products of the numbers of notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$. [i]S. Berlov[/i]

2007 Bulgaria Team Selection Test, 2
Tags: inequalities , number theory proposed , number theory
Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$

2007 India IMO Training Camp, 2
Tags: modular arithmetic , Diophantine equation , number theory proposed , number theory
Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$

2000 Romania Team Selection Test, 1
Tags: modular arithmetic , number theory proposed , number theory
Let $a>1$ be an odd positive integer. Find the least positive integer $n$ such that $2^{2000}$ is a divisor of $a^n-1$. [i]Mircea Becheanu [/i]

2011 Switzerland - Final Round, 3
Tags: number theory proposed , number theory
For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$. [i](Swiss Mathematical Olympiad, Final round, problem 3)[/i]