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: 1362

2010 Indonesia MO, 4
Tags: number theory unsolved , number theory
Given that $m$ and $n$ are positive integers with property: \[(mn)\mid(m^{2010}+n^{2010}+n)\] Show that there exists a positive integer $k$ such that $n=k^{2010}$ [i]Nanang Susyanto, Yogyakarta[/i]

2006 Italy TST, 2
Tags: modular arithmetic , Euler , number theory unsolved , number theory
Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$. a) Find all $n$ such that $A_{n}\neq \emptyset$ b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero. c) Is there $n$ such that $|{A_{n}}| = 130$?

1991 China Team Selection Test, 2
Tags: function , number theory unsolved , number theory
Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions: (1) $f(0) = 0, f(1) = 1,$ (2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$ Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$

2001 Federal Math Competition of S&M, Problem 1
Tags: calculus , derivative , number theory unsolved , number theory
Solve in positive integers \[ x^y + y = y^x + x \]

2010 German National Olympiad, 4
Tags: number theory unsolved , number theory
Find all positive integer solutions for the equation $(3x+1)(3y+1)(3z+1)=34xyz$ Thx

2008 South africa National Olympiad, 1
Tags: floor function , number theory unsolved , number theory
Determine the number of positive divisors of $2008^8$ that are less than $2008^4$.

2012 South africa National Olympiad, 4
Tags: function , number theory unsolved , number theory
Let $p$ and $k$ be positive integers such that $p$ is prime and $k>1$. Prove that there is at most one pair $(x,y)$ of positive integers such that $x^k+px=y^k$.

1997 Brazil Team Selection Test, Problem 3
Tags: modular arithmetic , linear algebra , matrix , number theory unsolved , number theory
Find all positive integers $x>1, y$ and primes $p,q$ such that $p^{x}=2^{y}+q^{x}$

1970 IMO Longlists, 27
Tags: search , number theory unsolved , number theory
Find a $n\in\mathbb{N}$ such that for all primes $p$, $n$ is divisible by $p$ if and only if $n$ is divisible by $p-1$.

2010 Tournament Of Towns, 1
Tags: number theory unsolved , number theory
$2010$ ships deliver bananas, lemons and pineapples from South America to Russia. The total number of bananas on each ship equals the number of lemons on all other ships combined, while the total number of lemons on each ship equals the total number of pineapples on all other ships combined. Prove that the total number of fruits is a multiple of $31$.

2000 All-Russian Olympiad, 6
Tags: Euler , number theory unsolved , number theory
A perfect number, greater than $28$ is divisible by $7$. Prove that it is also divisible by $49$.

1999 Brazil Team Selection Test, Problem 5
Tags: quadratics , number theory unsolved , number theory
(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove that $n$ is a power of $2$; (b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such that $2^n-1$ divides $m^2 + 9$.

2000 Polish MO Finals, 3
Tags: number theory unsolved , number theory
The sequence $p_1, p_2, p_3, ...$ is defined as follows. $p_1$ and $p_2$ are primes. $p_n$ is the greatest prime divisor of $p_{n-1} + p_{n-2} + 2000$. Show that the sequence is bounded.

1996 Bundeswettbewerb Mathematik, 2
Tags: modular arithmetic , induction , number theory unsolved , number theory
Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$ \[x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}\] where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.

2000 All-Russian Olympiad, 6
Tags: number theory unsolved , number theory , easy problem
A perfect number, greater than $6$, is divisible by $3$. Prove that it is also divisible by $9$.

2012 Finnish National High School Mathematics Competition, 3
Tags: algebra , binomial theorem , number theory unsolved , number theory
Prove that for all integers $k\geq 2,$ the number $k^{k-1}-1$ is divisible by $(k-1)^2.$

1992 Vietnam Team Selection Test, 2
Tags: quadratics , invariant , symmetry , number theory unsolved , number theory
Find all pair of positive integers $(x, y)$ satisfying the equation \[x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.\]

2004 Bulgaria National Olympiad, 2
Tags: number theory unsolved , number theory
For any positive integer $n$ the sum $\displaystyle 1+\frac 12+ \cdots + \frac 1n$ is written in the form $\displaystyle \frac{P(n)}{Q(n)}$, where $P(n)$ and $Q(n)$ are relatively prime. a) Prove that $P(67)$ is not divisible by 3; b) Find all possible $n$, for which $P(n)$ is divisible by 3.

2010 Contests, 2
Tags: floor function , number theory unsolved , number theory
Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n \minus{} 2)!$.

2017 Kyrgyzstan Regional Olympiad, 2
Tags: Diophantine equation , number theory unsolved , number theory
$x^2 + 2y^2 = 1$ solve in integers.

2013 Baltic Way, 18
Tags: induction , inequalities , number theory unsolved , number theory
Find all pairs $(x,y)$ of integers such that $y^3-1=x^4+x^2$.

2009 Singapore Team Selection Test, 3
Tags: number theory unsolved , number theory
Determine the smallest positive integer $\ N $ such that there exists 6 distinct integers $\ a_1, a_2, a_3, a_4, a_5, a_6 > 0 $ satisfying: (i) $\ N = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 $ (ii) $\ N - a_i$ is a perfect square for $\ i = 1,2,3,4,5,6 $.

2004 China Team Selection Test, 3
Tags: number theory , relatively prime , number theory unsolved
Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m\minus{}m$ can not be divided by $ p$ for any integer $ n$.

2002 China Team Selection Test, 3
Tags: modular arithmetic , induction , number theory unsolved , number theory
Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.

1981 Austrian-Polish Competition, 1
Tags: number theory unsolved , number theory
Find the smallest $n$ for which we can find $15$ distinct elements $a_{1},a_{2},...,a_{15}$ of $\{16,17,...,n\}$ such that $a_{k}$ is a multiple of $k$.