Found problems: 1362
2013 Singapore MO Open, 1
Let $a_1$, $a_2$, ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the $2013$-th powers of the digits of $a_n$. Do there exist distinct positive integers $i$, $j$ such that $a_i=a_j$?
2011 Kazakhstan National Olympiad, 4
Prove that there are infinitely many natural numbers, the arithmetic mean and geometric mean of the divisors which are both integers.
1998 China Team Selection Test, 3
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.
2002 Czech and Slovak Olympiad III A, 1
Solve the system
\[(4x)_5+7y=14 \\ (2y)_5 -(3x)_7=74\]
in the domain of integers, where $(n)_k$ stands for the multiple of the number $k$ closest to the number $n$.
1995 Turkey Team Selection Test, 2
Let $n\in\mathbb{N}$ be given. Prove that the following two conditions are equivalent:
$\quad(\text{i})\: n|a^n-a$ for any positive integer $a$;
$\quad(\text{ii})\:$ For any prime divisor $p$ of $n$, $p^2 \nmid n$ and $p-1|n-1$.
2009 Finnish National High School Mathematics Competition, 4
We say that the set of step lengths $D\subset \mathbb{Z}_+=\{1,2,\ldots\}$ is [i]excellent[/i] if it has the following property: If we split the set of integers into two subsets $A$ and $\mathbb{Z}\setminus{A}$, at least other set contains element $a-d,a,a+d$ (i.e. $\{a-d,a,a+d\} \subset A$ or $\{a-d,a,a+d\}\in \mathbb{Z}\setminus A$ from some integer $a\in \mathbb{Z},d\in D$.) For example the set of one element $\{1\}$ is not excellent as the set of integer can be split into even and odd numbers, and neither of these contains three consecutive integer. Show that the set $\{1,2,3,4\}$ is excellent but it has no proper subset which is excellent.
2011 Croatia Team Selection Test, 4
We define the sequence $x_n$ so that
\[x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.\]
Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.
2016 Germany Team Selection Test, 2
The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]
2009 All-Russian Olympiad, 8
Let $ x$, $ y$ be two integers with $ 2\le x, y\le 100$. Prove that $ x^{2^n} \plus{} y^{2^n}$ is not a prime for some positive integer $ n$.
2013 Moldova Team Selection Test, 1
Let $A=20132013...2013$ be formed by joining $2013$, $165$ times. Prove that $2013^2 \mid A$.
1987 China National Olympiad, 6
Sum of $m$ pairwise different positive even numbers and $n$ pairwise different positive odd numbers is equal to $1987$. Find, with proof, the maximum value of $3m+4n$.
2009 Belarus Team Selection Test, 1
Denote by $\phi(n)$ for all $n\in\mathbb{N}$ the number of positive integer smaller than $n$ and relatively prime to $n$. Also, denote by $\omega(n)$ for all $n\in\mathbb{N}$ the number of prime divisors of $n$. Given that $\phi(n)|n-1$ and $\omega(n)\leq 3$. Prove that $n$ is a prime number.
1978 IMO Longlists, 43
If $p$ is a prime greater than $3$, show that at least one of the numbers
\[\frac{3}{p^2} , \frac{4}{p^2} , \cdots, \frac{p-2}{p^2}\]
is expressible in the form $\frac{1}{x} + \frac{1}{y}$, where $x$ and $y$ are positive integers.
2018 Latvia Baltic Way TST, P16
Call a natural number [i]simple[/i] if it is not divisible by any square of a prime number (in other words it is square-free).
Prove that there are infinitely many positive integers $n$ such that both $n$ and $n+1$ are [i]simple[/i].
2012 Indonesia TST, 4
Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$.
Russian TST 2015, P4
Let $p \geq 5$ be a prime number. Prove that there exists a positive integer $a < p-1$ such that neither of $a^{p-1}-1$ and $(a+1)^{p-1}-1$ is divisible by $p^{2}$ .
2004 Belarusian National Olympiad, 4
For a positive integer $A = \overline{a_n ...a_1a_0}$ with nonzero digits which are not all the same ($n \ge 0$), the numbers $A_k = \overline{a_{n-k}...a_1a_0a_n ...a_{n-k+1}}$ are obtained for $k = 1,2,...,n$ by cyclic permutations of its digits. Find all $A$ for which each of the $A_k$ is divisible by $A$.
2014 Contests, 1
Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$
2013 ELMO Shortlist, 3
Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer.
Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers.
[i]Proposed by Matthew Babbitt[/i]
2014 IberoAmerican, 1
For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that
\[s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k).\]
2011 Kazakhstan National Olympiad, 4
Prove that there are infinitely many natural numbers, the arithmetic mean and geometric mean of the divisors which are both integers.
2013 Argentina National Olympiad, 6
A positive integer $n$ is called [i]pretty[/i] if there exists two divisors $d_1,d_2$ of $n$ $(1\leq d_1,d_2\leq n)$ such that $d_2-d_1=d$ for each divisor $d$ of $n$ (where $1<d<n$).
Find the smallest pretty number larger than $401$ that is a multiple of $401$.
2006 China Western Mathematical Olympiad, 1
Let $S=\{n|n-1,n,n+1$ can be expressed as the sum of the square of two positive integers.$\}$. Prove that if $n$ in $S$, $n^{2}$ is also in $S$.
2008 Vietnam National Olympiad, 3
Let $ m \equal{} 2007^{2008}$, how many natural numbers n are there such that $ n < m$ and $ n(2n \plus{} 1)(5n \plus{} 2)$ is divisible by $ m$ (which means that $ m \mid n(2n \plus{} 1)(5n \plus{} 2)$) ?
2003 Iran MO (3rd Round), 9
Does there exist an infinite set $ S$ such that for every $ a, b \in S$ we have $ a^2 \plus{} b^2 \minus{} ab \mid (ab)^2$.