Found problems: 1766
2001 District Olympiad, 2
Let $x,y,z\in \mathbb{R}^*$ such that $xy,yz,zx\in \mathbb{Q}$.
a) Prove that $x^2+y^2+z^2$ is rational;
b) If $x^3+y^3+z^3$ is rational, prove that $x,y,z$ are rational.
[i]Marius Ghergu[/i]
2012 ITAMO, 2
Determine all positive integers that are equal to $300$ times the sum of their digits.
2012 Indonesia MO, 3
Let $n$ be a positive integer. Show that the equation \[\sqrt{x}+\sqrt{y}=\sqrt{n}\] have solution of pairs of positive integers $(x,y)$ if and only if $n$ is divisible by some perfect square greater than $1$.
[i]Proposer: Nanang Susyanto[/i]
1977 IMO Longlists, 49
Find all pairs of integers $(p,q)$ for which all roots of the trinomials $x^2+px+q$ and $x^2+qx+p$ are integers.
2005 Irish Math Olympiad, 1
Show that $ 2005^{2005}$ is a sum of two perfect squares, but not a sum of two perfect cubes.
2006 Croatia Team Selection Test, 4
Find all natural solutions of $3^{x}= 2^{x}y+1.$
1997 Turkey MO (2nd round), 1
Find all pairs of integers $(x, y)$ such that $5x^{2}-6xy+7y^{2}=383$.
2003 Italy TST, 1
Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$
2011 Baltic Way, 18
Determine all pairs $(p,q)$ of primes for which both $p^2+q^3$ and $q^2+p^3$ are perfect squares.
2012 Puerto Rico Team Selection Test, 4
Let $a, b, c, d$ be digits such that $d > c > b > a \geq 0$. How many numbers of the form $1a1b1c1d1$ are
multiples of $33$?
2004 Iran MO (3rd Round), 18
Prove that for any $ n$, there is a subset $ \{a_1,\dots,a_n\}$ of $ \mathbb N$ such that for each subset $ S$ of $ \{1,\dots,n\}$, $ \sum_{i\in S}a_i$ has the same set of prime divisors.
2014 China Team Selection Test, 1
Prove that for any positive integers $k$ and $N$, \[\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},\] where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$).
Note: For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$.
2014 Contests, 1
Find the triplets of primes $(a,\ b,\ c)$ such that $a-b-8$ and $b-c-8$ are primes.
2010 Indonesia TST, 3
Let $ x$, $ y$, and $ z$ be integers satisfying the equation \[ \dfrac{2008}{41y^2}\equal{}\dfrac{2z}{2009}\plus{}\dfrac{2007}{2x^2}.\] Determine the greatest value that $ z$ can take.
[i]Budi Surodjo, Jogjakarta[/i]
2011 Baltic Way, 16
Let $a$ be any integer. Define the sequence $x_0,x_1,\ldots$ by $x_0=a$, $x_1=3$, and for all $n>1$
\[x_n=2x_{n-1}-4x_{n-2}+3.\]
Determine the largest integer $k_a$ for which there exists a prime $p$ such that $p^{k_a}$ divides $x_{2011}-1$.
2014 Turkey Team Selection Test, 2
$a_1=-5$, $a_2=-6$ and for all $n \geq 2$ the ${(a_n)^\infty}_{n=1}$ sequence defined as,
\[a_{n+1}=a_n+(a_1+1)(2a_2+1)(3a_3+1)\cdots((n-1)a_{n-1}+1)((n^2+n)a_n+2n+1)).\]
If a prime $p$ divides $na_n+1$ for a natural number n, prove that there is a integer $m$ such that $m^2\equiv5(modp)$
1975 Miklós Schweitzer, 4
Prove that the set of rational-valued, multiplicative arithmetical functions and the set of complex rational-valued, multiplicative arithmetical functions form isomorphic groups with the convolution operation $ f \circ g$ defined by \[{ (f \circ g)(n)= %Error. "displatmath" is a bad command.
\sum_{d|n} f(d)g(\frac nd}).\] (We call a complex number $ \textit{complex rational}$, if its real and imaginary parts are both rational.)
[i]B. Csakany[/i]
2003 Bulgaria Team Selection Test, 6
In natural numbers $m,n$ Solve : $n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4$
2008 Junior Balkan Team Selection Tests - Romania, 4
Let $ a,b$ be real nonzero numbers, such that number $ \lfloor an \plus{} b \rfloor$ is an even integer for every $ n \in \mathbb{N}$. Prove that $ a$ is an even integer.
1995 Irish Math Olympiad, 5
For each integer $ n$ of the form $ n\equal{}p_1 p_2 p_3 p_4$, where $ p_1,p_2,p_3,p_4$ are distinct primes, let $ 1\equal{}d_1<d_2<...<d_{15}<d_{16}\equal{}n$ be the divisors of $ n$. Prove that if $ n<1995$, then $ d_9\minus{}d_8 \not\equal{} 22$.
2000 JBMO ShortLists, 10
Prove that there are no integers $x,y,z$ such that
\[x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 \]
2002 Tournament Of Towns, 1
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
2013 Hong kong National Olympiad, 2
For any positive integer $a$, define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$. Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$.
2010 Kazakhstan National Olympiad, 5
Let $n \geq 2$ be an integer. Define $x_i =1$ or $-1$ for every $i=1,2,3,\cdots, n$.
Call an operation [i]adhesion[/i], if it changes the string $(x_1,x_2,\cdots,x_n)$ to $(x_1x_2, x_2x_3, \cdots ,x_{n-1}x_n, x_nx_1)$ .
Find all integers $n \geq 2$ such that the string $(x_1,x_2,\cdots, x_n)$ changes to $(1,1,\cdots,1)$ after finitely [i]adhesion[/i] operations.
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 $k\in\{m,m+1,\ldots,n\}$.