Found problems: 1766
2010 Federal Competition For Advanced Students, Part 1, 1
Let $f(n)=\sum_{k=0}^{2010}n^k$. Show that for any integer $m$ satisfying $2\leqslant m\leqslant 2010$, there exists no natural number $n$ such that $f(n)$ is divisible by $m$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 1)[/i]
2013 Olympic Revenge, 4
Find all triples $(p,n,k)$ of positive integers, where $p$ is a Fermat's Prime, satisfying \[p^n + n = (n+1)^k\].
[i]Observation: a Fermat's Prime is a prime number of the form $2^{\alpha} + 1$, for $\alpha$ positive integer.[/i]
1992 Taiwan National Olympiad, 4
For a positive integer number $r$, the sequence $a_{1},a_{2},...$ defined by $a_{1}=1$ and $a_{n+1}=\frac{na_{n}+2(n+1)^{2r}}{n+2}\forall n\geq 1$. Prove that each $a_{n}$ is positive integer number, and find $n's$ for which $a_{n}$ is even.
1996 Taiwan National Olympiad, 6
Let $q_{0},q_{1},...$ be a sequence of integers such that
a) for any $m>n$ we have $m-n\mid q_{m}-q_{n}$, and
b) $|q_{n}|\leq n^{10}, \ \forall n\geq 0$.
Prove there exists a polynomial $Q$ such that $q_{n}=Q(n), \ \forall n\geq 0$.
2006 Indonesia MO, 8
Find the largest $ 85$-digit integer which has property: the sum of its digits equals to the product of its digits.
2010 Contests, 2
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.
2004 Czech and Slovak Olympiad III A, 4
Find all positive integers $n$ such that $\sum_{k=1}^{n}\frac{n}{k!}$ is an integer.
1998 Taiwan National Olympiad, 1
Let $m,n$ are positive integers.
a)Prove that $(m,n)=2\sum_{k=0}^{m-1}[\frac{kn}{m}]+m+n-mn$.
b)If $m,n\geq 2$, prove that $\sum_{k=0}^{m-1}[\frac{kn}{m}]=\sum_{k=0}^{n-1}[\frac{km}{n}]$.
2014 Contests, 2
For the integer $n>1$, define $D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}$. Prove that for any integer $k>1$, there exists pairwise distinct positive integers $n_1,n_2,\ldots,n_k$ such that $n_1,\ldots,n_k>1$ and $|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2$.
2011 ELMO Shortlist, 3
Let $n>1$ be a fixed positive integer, and call an $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers greater than $1$ [i]good[/i] if and only if $a_i\Big|\left(\frac{a_1a_2\cdots a_n}{a_i}-1\right)$ for $i=1,2,\ldots,n$. Prove that there are finitely many good $n$-tuples.
[i]Mitchell Lee.[/i]
2014 ELMO Shortlist, 1
Does there exist a strictly increasing infinite sequence of perfect squares $a_1, a_2, a_3, ...$ such that for all $k\in \mathbb{Z}^+$ we have that $13^k | a_k+1$?
[i]Proposed by Jesse Zhang[/i]
2010 Romania National Olympiad, 4
Consider the set $\mathcal{F}$ of functions $f:\mathbb{N}\to\mathbb{N}$ (where $\mathbb{N}$ is the set of non-negative integers) having the property that
\[f(a^2-b^2)=f(a)^2-f(b)^2,\ \text{for all }a,b\in\mathbb{N},\ a\ge b.\]
a) Determine the set $\{f(1)\mid f\in\mathcal{F}\}$.
b) Prove that $\mathcal{F}$ has exactly two elements.
[i]Nelu Chichirim[/i]
1949 Miklós Schweitzer, 3
Let $ p$ be an odd prime number and $ a_1,a_2,...,a_p$ and $ b_1,b_2,...,b_p$ two arbitrary permutations of the numbers $ 1,2,...,p$ . Show that the least positive residues modulo $ p$ of the numbers $ a_1b_1, a_2b_2,...,a_pb_p$ never form a permutation of the numbers $ 1,2,...,p$.
2013 Saint Petersburg Mathematical Olympiad, 4
Find all pairs $(p,q)$ of prime numbers, such that $2p-1$, $2q-1$, $2pq-1$ are perfect square.
F. Petrov, A. Smirnov
2009 Korea National Olympiad, 3
For all positive integer $ n \ge 2 $, prove that $ 2^n -1 $ can't be a divisor of $ 3^n -1 $.
2013 Iran Team Selection Test, 5
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]
2005 ISI B.Math Entrance Exam, 4
For a set $S$ we denote its cardinality by $|S|$. Let $e_1,e_2,\ldots,e_k$ be non-negative integers. Let $A_k$ (respectively $B_k$) be the set of all $k$-tuples $(f_1,f_2,\ldots,f_k)$ of integers such that $0\leq f_i\leq e_i$ for all $i$ and $\sum_{i=1}^k f_i$ is even (respectively odd). Show that $|A_k|-|B_k|=0 \textrm{ or } 1$.
1995 Baltic Way, 4
Josh is older than Fred. Josh notices that if he switches the two digits of his age (an integer), he gets Fred’s age. Moreover, the difference between the squares of their ages is a square of an integer. How old are Josh and Fred?
2009 Indonesia TST, 1
a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime?
b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?
1999 Mongolian Mathematical Olympiad, Problem 4
Maybe well known:
$p$ a prime number, $n$ an integer. Prove that $n$ divides $\phi(p^n-1)$ where $\phi(x)$ is the Euler function.
2002 China Western Mathematical Olympiad, 1
Find all positive integers $ n$ such that $ n^4\minus{}4n^3\plus{}22n^2\minus{}36n\plus{}18$ is a perfect square.
2009 Macedonia National Olympiad, 1
Find all natural numbers $x,y,z$ such that $a+2^x3^y=z^2$.
2009 Indonesia TST, 3
Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n \minus{} 1$ divides $ 2^{k(n \minus{} 1)! \plus{} k^n} \minus{} 1$ for all integers $ k > n$.
2014 Cono Sur Olympiad, 2
A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i].
a) Prove that there are infinite [i]non-charrua[/i] pairs.
b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].
1970 IMO Longlists, 49
For $n \in \mathbb N$, let $f(n)$ be the number of positive integers $k \leq n$ that do not contain the digit $9$. Does there exist a positive real number $p$ such that $\frac{f(n)}{n} \geq p$ for all positive integers $n$?