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

1990 Federal Competition For Advanced Students, P2, 5
Tags: number theory unsolved , number theory
Determine all rational numbers $ r$ such that all solutions of the equation: $ rx^2\plus{}(r\plus{}1)x\plus{}(r\minus{}1)\equal{}0$ are integers.

1999 Hong kong National Olympiad, 1
Tags: number theory unsolved , number theory
Find all positive rational numbers $r\not=1$ such that $r^{\frac{1}{r-1}}$ is rational.

1993 Turkey Team Selection Test, 1
Tags: modular arithmetic , arithmetic sequence , number theory unsolved , number theory
Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.

2014 Mexico National Olympiad, 6
Tags: inequalities , induction , probability , number theory unsolved , number theory
Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all values of $n$ such that $n + d(n) = d(n)^2$.

2001 Moldova Team Selection Test, 1
Tags: number theory unsolved , number theory
Let $n$ be a positive integer of the form $4k+1$, $k\in \mathbb N$ and $A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}$. Prove that there exist integers $x,y$ such that $x^n+y^n \in A$ and $x+y \notin A$.

1983 IMO Longlists, 57
Tags: number theory unsolved , number theory
In the system of base $n^2 + 1$ find a number $N$ with $n$ different digits such that: [b](i)[/b] $N$ is a multiple of $n$. Let $N = nN'.$ [b](ii)[/b] The number $N$ and $N'$ have the same number $n$ of different digits in base $n^2 + 1$, none of them being zero. [b] (iii)[/b] If $s(C)$ denotes the number in base $n^2 + 1$ obtained by applying the permutation $s$ to the $n$ digits of the number $C$, then for each permutation $s, s(N) = ns(N').$

2011 Postal Coaching, 5
Tags: modular arithmetic , number theory unsolved , number theory
Let $(a_n )_{n\ge 1}$ be a sequence of integers that satisfies \[a_n = a_{n-1} -\text{min}(a_{n-2} , a_{n-3} )\] for all $n \ge 4$. Prove that for every positive integer $k$, there is an $n$ such that $a_n$ is divisible by $3^k$ .

1996 Polish MO Finals, 1
Tags: trigonometry , number theory unsolved , number theory
Find all pairs $(n,r)$ with $n$ a positive integer and $r$ a real such that $2x^2+2x+1$ divides $(x+1)^n - r$.

2000 Turkey Team Selection Test, 3
Tags: number theory unsolved , number theory
Let $P(x)=x+1$ and $Q(x)=x^2+1.$ Consider all sequences $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k), Q(y_k))$ or $(Q(x_k),P(y_k))$ for each $k. $ We say that a positive integer $n$ is nice if $x_n=y_n$ holds in at least one of these sequences. Find all nice numbers.

2014 Saudi Arabia BMO TST, 1
Tags: function , number theory , prime numbers , number theory unsolved
A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.

2010 Postal Coaching, 5
Tags: search , number theory unsolved , number theory
Find the first integer $n > 1$ such that the average of $1^2 , 2^2 ,\cdots, n^2$ is itself a perfect square.

2008 Baltic Way, 11
Tags: number theory unsolved , number theory
Consider a subset $A$ of $84$ elements of the set $\{1,\,2,\,\dots,\,169\}$ such that no two elements in the set add up to $169$. Show that $A$ contains a perfect square.

2003 China Team Selection Test, 2
Tags: number theory unsolved , number theory
Let $x<y$ be positive integers and $P=\frac{x^3-y}{1+xy}$. Find all integer values that $P$ can take.

2003 China Team Selection Test, 3
Tags: function , number theory unsolved , number theory
Let $x_0+\sqrt{2003}y_0$ be the minimum positive integer root of Pell function $x^2-2003y^2=1$. Find all the positive integer solutions $(x,y)$ of the equation, such that $x_0$ is divisible by any prime factor of $x$.

2006 South africa National Olympiad, 1
Tags: number theory unsolved , number theory
Reduce the fraction \[\frac{2121212121210}{1121212121211}\] to its simplest form.

2001 Irish Math Olympiad, 1
Tags: modular arithmetic , number theory unsolved , number theory
Find the least positive integer $ a$ such that $ 2001$ divides $ 55^n\plus{}a \cdot 32^n$ for some odd $ n$.

2014 Saudi Arabia IMO TST, 1
Tags: number theory unsolved , number theory
Tarik and Sultan are playing the following game. Tarik thinks of a number that is greater than $100$. Then Sultan is telling a number greater than $1$. If Tarik’s number is divisible by Sultan’s number, Sultan wins, otherwise Tarik subtracts Sultan’s number from his number and Sultan tells his next number. Sultan is forbidden to repeat his numbers. If Tarik’s number becomes negative, Sultan loses. Does Sultan have a winning strategy?

1985 IMO Longlists, 13
Tags: LaTeX , number theory unsolved , number theory
Find the average of the quantity \[(a_1 - a_2)^2 + (a_2 - a_3)^2 +\cdots + (a_{n-1} -a_n)^2\] taken over all permutations $(a_1, a_2, \dots , a_n)$ of $(1, 2, \dots , n).$

2007 Estonia Math Open Senior Contests, 7
Tags: number theory unsolved , number theory
Does there exist a natural number $ n$ such that $ n>2$ and the sum of squares of some $ n$ consecutive integers is a perfect square?

2010 Indonesia MO, 6
Tags: number theory , relatively prime , number theory unsolved
Find all positive integers $n>1$ such that \[\tau(n)+\phi(n)=n+1\] Which in this case, $\tau(n)$ represents the amount of positive divisors of $n$, and $\phi(n)$ represents the amount of positive integers which are less than $n$ and relatively prime with $n$. [i]Raja Oktovin, Pekanbaru[/i]

2009 Argentina Team Selection Test, 4
Tags: number theory unsolved , number theory
Find all positive integers $ n$ such that $ 20^n \minus{} 13^n \minus{} 7^n$ is divisible by $ 309$.

1995 Romania Team Selection Test, 2
Tags: number theory unsolved , number theory
Find all positive integers $ x,y,z,t$ such that $ x,y,z$ are pairwise coprime and $ (x \plus{} y)(y \plus{} z)(z \plus{} x) \equal{} xyzt$.

1996 Greece National Olympiad, 3
Tags: pigeonhole principle , number theory unsolved , number theory
Prove that among $81$ natural numbers whose prime divisors are in the set $\{2, 3, 5\}$ there exist four numbers whose product is the fourth power of an integer.

1988 IMO Longlists, 67
Tags: number theory unsolved , number theory
For each positive integer $ k$ and $ n,$ let $ S_k(n)$ be the base $ k$ digit sum of $ n.$ Prove that there are at most two primes $ p$ less than $20,000$ for which $ S_{31}(p)$ are composite numbers with at least two distinct prime divisors.

1983 Austrian-Polish Competition, 2
Tags: number theory unsolved , number theory
Find all triples of positive integers $(p, q, n)$ with $p$ and $q$ prime, such that $p(p+1)+q(q+1) = n(n+1)$.