Found problems: 1766
2013 Korea - Final Round, 5
Two coprime positive integers $ a, b $ are given. Integer sequence $ \{ a_n \}, \{b_n \} $ satisties
\[ (a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2 \]
Find all prime numbers $ p $ such that there exist positive integer $ n \le p $ satisfying $ p | b_n $.
2009 Hong Kong TST, 3
Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3}\plus{}4y^{4}\plus{}5z^{3}\minus{}y^{4}z \equiv 0$ is $ p^2$
1999 Iran MO (2nd round), 1
Find all positive integers $m$ such that there exist positive integers $a_1,a_2,\ldots,a_{1378}$ such that:
\[ m=\sum_{k=1}^{1378}{\frac{k}{a_k}}. \]
1994 Dutch Mathematical Olympiad, 3
$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes.
$ (b)$ Prove that every integer can be written as a sum of five cubes.
2011 All-Russian Olympiad, 3
Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$.
[i]A. Golovanov[/i]
2004 Iran MO (3rd Round), 17
Let $ p\equal{}4k\plus{}1$ be a prime. Prove that $ p$ has at least $ \frac{\phi(p\minus{}1)}2$ primitive roots.
2012 Indonesia TST, 4
Determine all integer $n > 1$ such that
\[\gcd \left( n, \dfrac{n-m}{\gcd(n,m)} \right) = 1\]
for all integer $1 \le m < n$.
2009 China Team Selection Test, 1
Let $ n$ be a composite. Prove that there exists positive integer $ m$ satisfying $ m|n, m\le\sqrt {n},$ and $ d(n)\le d^3(m).$ Where $ d(k)$ denotes the number of positive divisors of positive integer $ k.$
2011 All-Russian Olympiad Regional Round, 9.5
Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+4)$ is an integer. (Author: O. Podlipski)
2012 Spain Mathematical Olympiad, 1
Find all positive integers $n$ and $k$ such that $(n+1)^n=2n^k+3n+1$.
2014 Moldova Team Selection Test, 1
Prove that there do not exist $4$ points in the plane such that the distances between any pair of them is an odd integer.
2005 All-Russian Olympiad Regional Round, 9.7
9.7 Is there an infinite arithmetic sequence $\{a_n\}\subset \mathbb N$ s.t. $a_n+...+a_{n+9}\mid a_n...a_{n+9}$ for all $n$?
([i]V. Senderov[/i])
2012 ELMO Shortlist, 7
A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$).
Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime.
[i]Bobby Shen.[/i]
1977 IMO Longlists, 26
Let $p$ be a prime number greater than $5.$ Let $V$ be the collection of all positive integers $n$ that can be written in the form $n = kp + 1$ or $n = kp - 1 \ (k = 1, 2, \ldots).$ A number $n \in V$ is called [i]indecomposable[/i] in $V$ if it is impossible to find $k, l \in V$ such that $n = kl.$ Prove that there exists a number $N \in V$ that can be factorized into indecomposable factors in $V$ in more than one way.
2011 Baltic Way, 17
Determine all positive integers $d$ such that whenever $d$ divides a positive integer $n$, $d$ will also divide any integer obtained by rearranging the digits of $n$.
2014 Dutch BxMO/EGMO TST, 4
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 $k\in\{m,m+1,\ldots,n\}$.
2016 SGMO, Q1
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for any pair of naturals $m,n$,
$$\gcd(f(m),n) = \gcd(m,f(n)).$$
2007 Irish Math Olympiad, 1
Find all prime numbers $ p$ and $ q$ such that $ p$ divides $ q\plus{}6$ and $ q$ divides $ p\plus{}6$.
2008 APMO, 5
Let $ a, b, c$ be integers satisfying $ 0 < a < c \minus{} 1$ and $ 1 < b < c$. For each $ k$, $ 0\leq k \leq a$, Let $ r_k,0 \leq r_k < c$
be the remainder of $ kb$ when divided by $ c$. Prove that the two sets $ \{r_0, r_1, r_2, \cdots , r_a\}$ and $ \{0, 1, 2, \cdots , a\}$ are different.
2010 ELMO Shortlist, 3
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that
\begin{align*}
a^2 + b^2 + 3 &= 4ab\\
c^2 + d^2 + 3 &= 4cd\\
4c^3 - 3c &= a
\end{align*}
[i]Travis Hance.[/i]
2014 China Team Selection Test, 2
Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying:
(1)$\tau (n)=a$
(2)$n|\phi (n)+\sigma (n)$
Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.
2011 Serbia National Math Olympiad, 2
Let $n$ be an odd positive integer such that both $\phi(n)$ and $\phi (n+1)$ are powers of two. Prove $n+1$ is power of two or $n=5$.
2012 Middle European Mathematical Olympiad, 4
The sequence $ \{ a_n \} _ { n \ge 0 } $ is defined by $ a_0 = 2 , a_1 = 4 $ and
\[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \]
for all positive integers $ n $. Determine all prime numbers $ p $ for which there exists a positive integer $ m $ such that $ p $ divides the number $ a_m - 1 $.
2008 Spain Mathematical Olympiad, 1
Find two positive integers $a$ and $b$, when their sum and their least common multiple is given. Find the numbers when the sum is $3972$ and the least common multiple is $985928$.
2008 Czech-Polish-Slovak Match, 3
Find all primes $p$ such that the expression
\[\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2\]
is divisible by $p^3$.