Found problems: 1766
1982 IMO Longlists, 1
[b](a)[/b] Prove that $\frac{1}{n+1} \cdot \binom{2n}{n}$ is an integer for $n \geq 0.$
[b](b)[/b] Given a positive integer $k$, determine the smallest integer $C_k$ with the property that $\frac{C_k}{n+k+1} \cdot \binom{2n}{n}$ is an integer for all $n \geq k.$
2002 ITAMO, 5
Prove that if $m=5^n+3^n+1$ is a prime, then $12$ divides $n$.
1996 Baltic Way, 9
Let $n$ and $k$ be integers, $1\le k\le n$. Find an integer $b$ and a set $A$ of $n$ integers satisfying the following conditions:
(i) No product of $k-1$ distinct elements of $A$ is divisible by $b$.
(ii) Every product of $k$ distinct elements of $A$ is divisible by $b$.
(iii) For all distinct $a,a'$ in $A$, $a$ does not divide $a'$.
2005 ITAMO, 2
Prove that among any $18$ consecutive positive integers not exceeding $2005$ there is at least one divisible by the sum of its digits.
1990 IMO Longlists, 38
Let $\alpha$ be the positive root of the quadratic equation $x^2 = 1990x + 1$. For any $m, n \in \mathbb N$, define the operation $m*n = mn + [\alpha m][ \alpha n]$, where $[x]$ is the largest integer no larger than $x$. Prove that $(p*q)*r = p*(q*r)$ holds for all $p, q, r \in \mathbb N.$
2010 ITAMO, 6
Prove that there are infinitely many prime numbers that divide at least one integer of the form $2^{n^3+1}-3^{n^2+1}+5^{n+1}$ where $n$ is a positive integer.
Oliforum Contest II 2009, 4
Let $ m$ a positive integer and $ p$ a prime number, both fixed. Define $ S$ the set of all $ m$-uple of positive integers $ \vec{v} \equal{} (v_1,v_2,\ldots,v_m)$ such that $ 1 \le v_i \le p$ for all $ 1 \le i \le m$. Define also the function $ f(\cdot): \mathbb{N}^m \to \mathbb{N}$, that associates every $ m$-upla of non negative integers $ (a_1,a_2,\ldots,a_m)$ to the integer $ \displaystyle f(a_1,a_2,\ldots,a_m) \equal{} \sum_{\vec{v} \in S} \left(\prod_{1 \le i \le m}{v_i^{a_i}} \right)$.
Find all $ m$-uple of non negative integers $ (a_1,a_2,\ldots,a_m)$ such that $ p \mid f(a_1,a_2,\ldots,a_m)$.
[i](Pierfrancesco Carlucci)[/i]
2010 Switzerland - Final Round, 3
For $ n\in\mathbb{N}$, determine the number of natural solutions $ (a,b)$ such that
\[ (4a\minus{}b)(4b\minus{}a)\equal{}2010^n\]
holds.
2010 Slovenia National Olympiad, 1
Find all prime numbers $p, q$ and $r$ such that $p>q>r$ and the numbers $p-q, p-r$ and $q-r$ are also prime.
2014 ELMO Shortlist, 3
Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$.
[i]Proposed by Michael Kural[/i]
1997 Iran MO (2nd round), 1
Let $x,y$ be positive integers such that $3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.
2013 Korea Junior Math Olympiad, 6
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying
\[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \]
for all positive integer $m,n$.
2002 Junior Balkan MO, 3
Find all positive integers which have exactly 16 positive divisors $1 = d_1 < d_2 < \ldots < d_{16} =n$ such that the divisor $d_k$, where $k = d_5$, equals $(d_2 + d_4) d_6$.
2011 ELMO Shortlist, 4
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and
\[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\]
[i]Alex Zhu.[/i]
[hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]
2012 Middle European Mathematical Olympiad, 8
For any positive integer $n $ let $ d(n) $ denote the number of positive divisors of $ n $. Do there exist positive integers $ a $ and $b $, such that $ d(a)=d(b)$ and $ d(a^2 ) = d(b^2 ) $, but $ d(a^3 ) \ne d(b^3 ) $ ?
2014 China Girls Math Olympiad, 5
Let $a$ be a positive integer, but not a perfect square; $r$ is a real root of the equation $x^3-2ax+1=0$. Prove that $ r+\sqrt{a}$ is an irrational number.
2004 Italy TST, 2
A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$.
$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.
$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
2002 India IMO Training Camp, 8
Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$.
[list]
[b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$
[b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]
2014 Dutch BxMO/EGMO TST, 1
Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$
1990 Turkey Team Selection Test, 5
Let $b_m$ be numbers of factors $2$ of the number $m!$ (that is, $2^{b_m}|m!$ and $2^{b_m+1}\nmid m!$). Find the least $m$ such that $m-b_m = 1990$.
2002 Greece National Olympiad, 4
(a) Positive integers $p,q,r,a$ satisfy $pq=ra^2$, where $r$ is prime and $p,q$ are relatively prime. Prove that one of the numbers $p,q$ is a perfect square.
(b) Examine if there exists a prime $p$ such that $p(2^{p+1}-1)$ is a perfect square.
2009 Indonesia TST, 1
Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.
2023 Dutch BxMO TST, 5
Find all pairs of prime numbers $(p,q)$ for which
\[2^p = 2^{q-2} + q!.\]
2010 Postal Coaching, 2
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
2012 Pan African, 2
Find all positive integers $m$ and $n$ such that $n^m - m$ divides $m^2 + 2m$.