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

2014 Grand Duchy of Lithuania, 4
Tags: number theory proposed , number theory
Determine all positive integers $n > 1$ for which $n + D(n)$ is a power of $10$, where $D(n)$ denotes the largest integer divisor of $n$ satisfying $D(n) < n$.

2005 Germany Team Selection Test, 3
Tags: number theory proposed , number theory
Let $a$, $b$, $c$, $d$ and $n$ be positive integers such that $7\cdot 4^n = a^2+b^2+c^2+d^2$. Prove that the numbers $a$, $b$, $c$, $d$ are all $\geq 2^{n-1}$.

2002 France Team Selection Test, 3
Tags: number theory proposed , number theory
Let $p\ge 3$ be a prime number. Show that there exist $p$ positive integers $a_1,a_2,\ldots ,a_p$ not exceeding $2p^2$ such that the $\frac{p(p-1)}{2}$ sums $a_i+a_j\ (i<j)$ are all distinct.

2009 Iran Team Selection Test, 4
Tags: algebra , polynomial , number theory proposed , number theory
Find all polynomials $f$ with integer coefficient such that, for every prime $p$ and natural numbers $u$ and $v$ with the condition: \[ p \mid uv - 1 \] we always have $p \mid f(u)f(v) - 1$.

2009 JBMO Shortlist, 1
Tags: number theory proposed , number theory
Solve in non-negative integers the equation $ 2^{a}3^{b} \plus{} 9 \equal{} c^{2}$

2013 Stars Of Mathematics, 3
Tags: modular arithmetic , quadratics , number theory proposed , number theory
Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$. i) Prove there exist infinitely many primes, none dividing any term of the sequence. ii) Prove there exist infinitely many primes, each dividing some term of the sequence. [i](Dan Schwarz)[/i]

1971 IMO Longlists, 44
Tags: induction , logarithms , number theory proposed , number theory
Let $m$ and $n$ denote integers greater than $1$, and let $\nu (n)$ be the number of primes less than or equal to $n$. Show that if the equation $\frac{n}{\nu(n)}=m$ has a solution, then so does the equation $\frac{n}{\nu(n)}=m-1$.

2001 Macedonia National Olympiad, 1
Tags: modular arithmetic , number theory , greatest common divisor , number theory proposed
Prove that if $m$ and $s$ are integers with $ms=2000^{2001}$, then the equation $mx^2-sy^2=3$ has no integer solutions.

1990 Romania Team Selection Test, 10
Tags: number theory , prime numbers , number theory proposed
Let $p,q$ be positive prime numbers and suppose $q>5$. Prove that if $q \mid 2^{p}+3^{p}$, then $q>p$. [i]Laurentiu Panaitopol[/i]

2011 Iran Team Selection Test, 12
Tags: function , pigeonhole principle , induction , modular arithmetic , number theory proposed , number theory
Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

2016 Tuymaada Olympiad, 5
Tags: number theory , number theory proposed , Tuymaada , prime numbers
The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$). Prove that there are two consecutive positive integers such that the largest prime divisor of one of them is $p$ and that of the other is $q$.

1991 IberoAmerican, 4
Tags: number theory proposed , number theory
Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$.

2010 Iran MO (3rd Round), 5
Tags: number theory proposed , number theory
prove that if $p$ is a prime number such that $p=12k+\{2,3,5,7,8,11\}$($k \in \mathbb N \cup \{0\}$), there exist a field with $p^2$ elements.($\frac{100}{6}$ points)

2012 Turkey Team Selection Test, 3
Tags: modular arithmetic , number theory , prime numbers , number theory proposed
Let $\mathbb{Z^+}$ and $\mathbb{P}$ denote the set of positive integers and the set of prime numbers, respectively. A set $A$ is called $S-\text{proper}$ where $A, S \subset \mathbb{Z^+}$ if there exists a positive integer $N$ such that for all $a \in A$ and for all $0 \leq b <a$ there exist $s_1, s_2, \ldots, s_n \in S$ satisfying $ b \equiv s_1+s_2+\cdots+s_n \pmod a$ and $1 \leq n \leq N.$ Find a subset $S$ of $\mathbb{Z^+}$ for which $\mathbb{P}$ is $S-\text{proper}$ but $\mathbb{Z^+}$ is not.

1991 Iran MO (2nd round), 1
Tags: number theory proposed , number theory
Prove that the equation $x+x^2=y+y^2+y^3$ do not have any solutions in positive integers.

2005 Iran MO (3rd Round), 2
Tags: number theory proposed , number theory
Let $a\in\mathbb N$ and $m=a^2+a+1$. Find the number of $0\leq x\leq m$ that:\[x^3\equiv1(\mbox{mod}\ m)\]

2010 China Girls Math Olympiad, 3
Tags: modular arithmetic , number theory , relatively prime , number theory proposed
Prove that for every given positive integer $n$, there exists a prime $p$ and an integer $m$ such that $(a)$ $p \equiv 5 \pmod 6$ $(b)$ $p \nmid n$ $(c)$ $n \equiv m^3 \pmod p$

1993 Iran MO (2nd round), 3
Tags: pigeonhole principle , number theory proposed , number theory
Let $n, r$ be positive integers. Find the smallest positive integer $m$ satisfying the following condition. For each partition of the set $\{1, 2, \ldots ,m \}$ into $r$ subsets $A_1,A_2, \ldots ,A_r$, there exist two numbers $a$ and $b$ in some $A_i, 1 \leq i \leq r$, such that \[ 1 < \frac ab < 1 +\frac 1n.\]

2002 Hungary-Israel Binational, 1
Tags: number theory proposed , number theory
Find the greatest exponent $k$ for which $2001^{k}$ divides $2000^{2001^{2002}}+2002^{2001^{2000}}$.

2005 Indonesia MO, 6
Tags: symmetry , geometry , inequalities , number theory proposed , number theory
Find all triples $ (x,y,z)$ of integers which satisfy $ x(y \plus{} z) \equal{} y^2 \plus{} z^2 \minus{} 2$ $ y(z \plus{} x) \equal{} z^2 \plus{} x^2 \minus{} 2$ $ z(x \plus{} y) \equal{} x^2 \plus{} y^2 \minus{} 2$.

1986 IMO Longlists, 32
Tags: number theory proposed , number theory
Find, with proof, all solutions of the equation $\frac 1x +\frac 2y- \frac 3z = 1$ in positive integers $x, y, z.$

2008 Moldova Team Selection Test, 1
Tags: number theory proposed , number theory
Let $ p$ be a prime number. Solve in $ \mathbb{N}_0\times\mathbb{N}_0$ the equation $ x^3\plus{}y^3\minus{}3xy\equal{}p\minus{}1$.

2014 German National Olympiad, 1
Tags: number theory , number theory proposed , prime numbers
For which non-negative integers $n$ is \[K=5^{2n+3} + 3^{n+3} \cdot 2^n\] prime?

2014 JBMO TST - Turkey, 3
Tags: number theory proposed , number theory
Find all pairs $(m, n)$ of positive integers satsifying $m^6+5n^2=m+n^3$.

2002 Taiwan National Olympiad, 1
Tags: number theory proposed , number theory
Find all natural numbers $n$ and nonnegative integers $x_{1},x_{2},...,x_{n}$ such that $\sum_{i=1}^{n}x_{i}^{2}=1+\frac{4}{4n+1}(\sum_{i=1}^{n}x_{i})^{2}$.