Found problems: 1362
2014 Postal Coaching, 5
Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.
2000 France Team Selection Test, 3
Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.
2005 Turkey MO (2nd round), 6
Suppose that a sequence $(a_n)_{n=1}^{\infty}$ of integers has the following property: For all $n$ large enough (i.e. $n \ge N$ for some $N$ ), $a_n$ equals the number of indices $i$, $1 \le i < n$, such that $a_i + i \ge n$. Find the maximum possible number of integers which occur infinitely many times in the sequence.
2004 China Second Round Olympiad, 3
For integer $n\ge 4$, find the minimal integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set ${m, m+1, \ldots, m+n+1}$ there are at least $3$ relatively prime elements.
2004 APMO, 4
For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that
\[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \]
is even for every positive integer $n$.
2006 China Team Selection Test, 2
Given positive integers $m$, $a$, $b$, $(a,b)=1$. $A$ is a non-empty subset of the set of all positive integers, so that for every positive integer $n$ there is $an \in A$ and $bn \in A$. For all $A$ that satisfy the above condition, find the minimum of the value of $\left| A \cap \{ 1,2, \cdots,m \} \right|$
2002 USA Team Selection Test, 6
Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$.
2010 Postal Coaching, 1
In a family there are four children of different ages, each age being a positive integer not less than $2$ and not greater than $16$. A year ago the square of the age of the eldest child was equal to the sum of the squares of the ages of the remaining children. One year from now the sum of the squares of the youngest and the oldest will be equal to the sum of the squares of the other two. How old is each child?
2011 Kazakhstan National Olympiad, 4
We write in order of increasing number of 1 and all positive integers,which the sum of digits is divisible by $5$. Obtain a sequence of $1, 5, 14, 19. . .$
Prove that the n-th term of the sequence is less than $5n$.
1993 Cono Sur Olympiad, 2
Prove that there exists a succession $a_1, a_2, ... , a_k, ...$, where each $a_i$ is a digit ($a_i \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)$ ) and $a_0=6$, such that, for each positive integrer $n$, the number $x_n=a_0+10a_1+100a_2+...+10^{n-1}a_{n-1}$ verify that $x_n^2-x_n$ is divisible by $10^n$.
2011 Korea National Olympiad, 1
Find the number of positive integer $ n < 3^8 $ satisfying the following condition.
"The number of positive integer $k (1 \leq k \leq \frac {n}{3})$ such that $ \frac{n!}{(n-3k)! \cdot k! \cdot 3^{k+1}} $ is not a integer" is $ 216 $.
1987 IberoAmerican, 3
Prove that if $m,n,r$ are positive integers, and:
\[1+m+n\sqrt{3}=(2+\sqrt{3})^{2r-1} \]
then $m$ is a perfect square.
1983 Canada National Olympiad, 4
Prove that for every prime number $p$, there are infinitely many positive integers $n$ such that $p$ divides $2^n - n$.
2011 Indonesia TST, 4
A prime number $p$ is a [b]moderate[/b] number if for every $2$ positive integers $k > 1$ and $m$, there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \]
If $q$ is the smallest [b]moderate[/b] number, then determine the smallest prime $r$ which is not moderate and $q < r$.
2013 Kazakhstan National Olympiad, 1
Find all triples of positive integer $(m,n,k)$ such that $ k^m|m^n-1$ and $ k^n|n^m-1$
1989 Kurschak Competition, 2
For any positive integer $n$ denote $S(n)$ the digital sum of $n$ when represented in the decimal system. Find every positive integer $M$ for which $S(Mk)=S(M)$ holds for all integers $1\le k\le M$.
2013 Kazakhstan National Olympiad, 1
On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.
1986 Canada National Olympiad, 4
For all positive integers $n$ and $k$, define $F(n,k) = \sum_{r = 1}^n r^{2k - 1}$. Prove that $F(n,1)$ divides $F(n,k)$.
2006 Finnish National High School Mathematics Competition, 1
Determine all pairs $(x, y)$ of positive integers for which the equation \[x + y + xy = 2006\] holds.
2010 Brazil National Olympiad, 3
Find all pairs $(a, b)$ of positive integers such that
\[ 3^a = 2b^2 + 1. \]
2006 Czech-Polish-Slovak Match, 4
Show that for every integer $k \ge 1$ there is a positive integer $n$ such that the decimal representation of $2^n$ contains a block of exactly $k$ zeros, i.e. $2^n = \dots a00 \dots 0b \cdots$ with $k$ zeros and $a, b \ne 0$.
2014 Czech and Slovak Olympiad III A, 1
Let be $n$ a positive integer. Denote all its (positive) divisors as $1=d_1<d_2<\cdots<d_{k-1}<d_k=n$.
Find all values of $n$ satisfying $d_5-d_3=50$ and $11d_5+8d_7=3n$.
(Day 1, 1st problem
author: Matúš Harminc)
1980 IMO, 18
Do there exist $\{x,y\}\in\mathbb{Z}$ satisfying $(2x+1)^{3}+1=y^{4}$?
2011 USA Team Selection Test, 6
A polynomial $P(x)$ is called [i]nice[/i] if $P(0) = 1$ and the nonzero coefficients of $P(x)$ alternate between $1$ and $-1$ when written in order. Suppose that $P(x)$ is nice, and let $m$ and $n$ be two relatively prime positive integers. Show that
\[Q(x) = P(x^n) \cdot \frac{(x^{mn} - 1)(x-1)}{(x^m-1)(x^n-1)}\]
is nice as well.
1998 IberoAmerican, 3
Find the minimum natural number $n$ with the following property: between any collection of $n$ distinct natural numbers in the set $\{1,2, \dots,999\}$ it is possible to choose four different $a,\ b,\ c,\ d$ such that: $a + 2b + 3c = d$.