Found problems: 1362
2010 Germany Team Selection Test, 3
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$
2010 Costa Rica - Final Round, 4
Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]
2001 Bulgaria National Olympiad, 1
Consider the sequence $\{a_n\}$ such that $a_0=4$, $a_1=22$, and $a_n-6a_{n-1}+a_{n-2}=0$ for $n\ge2$. Prove that there exist sequences $\{x_n\}$ and $\{y_n\}$ of positive integers such that
\[
a_n=\frac{y_n^2+7}{x_n-y_n}
\]
for any $n\ge0$.
1988 IMO Longlists, 64
Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$
2010 China Team Selection Test, 3
Fine all positive integers $m,n\geq 2$, such that
(1) $m+1$ is a prime number of type $4k-1$;
(2) there is a (positive) prime number $p$ and nonnegative integer $a$, such that
\[\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\]
2010 Contests, 4
With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?
2014 Iran MO (3rd Round), 3
(a) $n$ is a natural number. $d_1,\dots,d_n,r_1,\dots ,r_n$ are natural numbers such that for each $i,j$ that $1\leq i < j \leq n$ we have $(d_i,d_j)=1$ and $d_i\geq 2$.
Prove that there exist an $x$ such that
(i) $1 \leq x \leq 3^n$
(ii)For each $1 \leq i \leq n$ \[x \overset{d_i}{\not{\equiv}} r_i\]
(b) For each $\epsilon >0$ prove that there exists natural $N$ such that for each $n>N$ and each $d_1,\dots,d_n,r_1,\dots ,r_n$ satisfying the conditions above there exists an $x$ satisfying (ii) such that $1\leq x \leq (2+\epsilon )^n$.
Time allowed for this exam was 75 minutes.
2011 Nordic, 1
When $a_0, a_1, \dots , a_{1000}$ denote digits, can the sum of the $1001$-digit numbers $a_0a_1\cdots a_{1000}$ and $a_{1000}a_{999}\cdots a_0$ have odd digits only?
1995 Hungary-Israel Binational, 1
Let the sum of the first $ n$ primes be denoted by $ S_n$. Prove that for any positive integer $ n$, there exists a perfect square between $ S_n$ and $ S_{n\plus{}1}$.
2001 Finnish National High School Mathematics Competition, 5
Determine $n \in \Bbb{N}$ such that $n^2 + 2$ divides $2 + 2001n.$
2000 Bulgaria National Olympiad, 3
Let $ p$ be a prime number and let $ a_1,a_2,\ldots,a_{p \minus{} 2}$ be positive integers such that $ p$ doesn't $ a_k$ or $ {a_k}^k \minus{} 1$ for any $ k$. Prove that the product of some of the $ a_i$'s is congruent to $ 2$ modulo $ p$.
1992 China Team Selection Test, 3
For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$
1996 Baltic Way, 6
Let $a,b,c,d$ be positive integers such that $ab\equal{}cd$. Prove that $a\plus{}b\plus{}c\plus{}d$ is a composite number.
2011 India IMO Training Camp, 3
Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that
\[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\]
for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that
\[a_{k+2011}=a_{k+2011^{2011}}.\]
1998 Vietnam Team Selection Test, 2
Let $d$ be a positive divisor of $5 + 1998^{1998}$. Prove that $d = 2 \cdot x^2 + 2 \cdot x \cdot y + 3 \cdot y^2$, where $x, y$ are integers if and only if $d$ is congruent to 3 or 7 $\pmod{20}$.
1990 China National Olympiad, 4
Given a positive integer number $a$ and two real numbers $A$ and $B$, find a necessary and sufficient condition on $A$ and $B$ for the following system of equations to have integer solution:
\[ \left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2\\ x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right. \]
2014 Bundeswettbewerb Mathematik, 4
Find all postive integers $n$ for which the number $\frac{4n+1}{n(2n-1)}$ has a terminating decimal expansion.
2013 Polish MO Finals, 2
There are given integers $a$ and $b$ such that $a$ is different from $0$ and the number $3+ a +b^2$ is divisible by $6a$. Prove that $a$ is negative.
2018 Latvia Baltic Way TST, P14
Let $a_1,a_2,...$ be a sequence of positive integers with $a_1=2$. For each $n \ge 1$, $a_{n+1}$ is the biggest prime divisor of $a_1a_2...a_n+1$.
Prove that the sequence does not contain numbers $5$ and $11$.
2010 IMC, 4
Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.
2002 Germany Team Selection Test, 3
Determine all $(x,y) \in \mathbb{N}^2$ which satisfy $x^{2y} + (x+1)^{2y} = (x+2)^{2y}.$
2017 Ukraine Team Selection Test, 12
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
2005 All-Russian Olympiad, 4
Integers $x>2,\,y>1,\,z>0$ satisfy an equation $x^y+1=z^2$. Let $p$ be a number of different prime divisors of $x$, $q$ be a number of different prime divisors of $y$. Prove that $p\geq q+2$.
2005 Postal Coaching, 21
Find all positive integers $n$ that can be [i]uniquely[/i] expressed as a sum of five or fewer squares.
2008 Tuymaada Olympiad, 8
250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t$ is divisible by 23.
[i]Author: K. Kokhas[/i]