Found problems: 1766
2010 Indonesia TST, 3
Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows:
(1). $ \dfrac{1}{2} \in \mathbb{H}$,
(2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1\plus{}x} \in \mathbb{H}$ and also $ \dfrac{x}{1\plus{}x} \in \mathbb{H}$.
Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.
2015 German National Olympiad, 4
Let $k$ be a positive integer. Define $n_k$ to be the number with decimal representation $70...01$ where there are exactly $k$ zeroes. Prove the following assertions:
a) None of the numbers $n_k$ is divisible by $13$.
b) Infinitely many of the numbers $n_k$ are divisible by $17$.
2010 Tournament Of Towns, 1
In a multiplication table, the entry in the $i$-th row and the $j$-th column is the product $ij$ From an $m\times n$ subtable with both $m$ and $n$ odd, the interior $(m-2) (n-2)$ rectangle is removed, leaving behind a frame of width $1$. The squares of the frame are painted alternately black and white. Prove that the sum of the numbers in the black squares is equal to the sum of the numbers in the white squares.
2012 Romania Team Selection Test, 3
Let $a_1$ , $\ldots$ , $a_n$ be positive integers and $a$ a positive integer that is greater than $1$ and is divisible by the product $a_1a_2\ldots a_n$. Prove that $a^{n+1}+a-1$ is not divisible by the product $(a+a_1-1)(a+a_2-1)\ldots(a+a_n-1)$.
2007 Moldova Team Selection Test, 2
Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.
2002 Tournament Of Towns, 3
In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
1997 Taiwan National Olympiad, 4
Let $k=2^{2^{n}}+1$ for some $n\in\mathbb{N}$. Show that $k$ is prime iff $k|3^{\frac{k-1}{2}}+1$.
1987 Bundeswettbewerb Mathematik, 1
Find all non-negative integer solutions of the equation
\[2^x + 3^y = z^2 .\]
2019 IMEO, 5
Find all pairs of positive integers $(s, t)$, so that for any two different positive integers $a$ and $b$ there exists some positive integer $n$, for which $$a^s + b^t | a^n + b^{n+1}.$$
[i]Proposed by Oleksii Masalitin (Ukraine)[/i]
1990 Iran MO (2nd round), 3
[b](a)[/b] For every positive integer $n$ prove that
\[1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} <2\]
[b](b)[/b] Let $X=\{1, 2, 3 ,\ldots, n\} \ ( n \geq 1)$ and let $A_k$ be non-empty subsets of $X \ (k=1,2,3, \ldots , 2^n -1).$ If $a_k$ be the product of all elements of the set $A_k,$ prove that
\[\sum_{i=1}^{m} \sum_{j=1}^m \frac{1}{a_i \cdot j^2} <2n+1\]
2010 Contests, 4
Let $a,b,c$ be given positive integers. Prove that there exists some positive integer $N$ such that
\[ a\mid Nbc+b+c,\ b\mid Nca+c+a,\ c\mid Nab+a+b \]
if and only if, denoting $d=\gcd(a,b,c)$ and $a=dx$, $b=dy$, $c=dz$, the positive integers $x,y,z$ are pairwise coprime, and also $\gcd(d,xyz) \mid x+y+z$.
(Dan Schwarz)
2007 Romania Team Selection Test, 1
Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.
2011 Olympic Revenge, 2
Let $p$ be a fixed prime. Determine all the integers $m$, as function of $p$, such that there exist $a_1, a_2, \ldots, a_p \in \mathbb{Z}$ satisfying
\[m \mid a_1^p + a_2^p + \cdots + a_p^p - (p+1).\]
1999 Turkey Team Selection Test, 1
Let $m \leq n$ be positive integers and $p$ be a prime. Let $p-$expansions of $m$ and $n$ be
\[m = a_0 + a_1p + \dots + a_rp^r\]\[n = b_0 + b_1p + \dots + b_sp^s\]
respectively, where $a_r, b_s \neq 0$, for all $i \in \{0,1,\dots,r\}$ and for all $j \in \{0,1,\dots,s\}$, we have $0 \leq a_i, b_j \leq p-1$ .
If $a_i \leq b_i$ for all $i \in \{0,1,\dots,r\}$, we write $ m \prec_p n$. Prove that
\[p \nmid {{n}\choose{m}} \Leftrightarrow m \prec_p n\].
2010 Contests, 4
Let $p$ be a prime number of the form $4k+3$. Prove that there are no integers $w,x,y,z$ whose product is not divisible by $p$, such that:
\[
w^{2p}+x^{2p}+y^{2p}=z^{2p}.
\]
1995 All-Russian Olympiad, 1
Can the numbers $1,2,3,\ldots,100$ be covered with $12$ geometric progressions?
[i]A. Golovanov[/i]
1993 Hungary-Israel Binational, 1
Find all pairs of coprime natural numbers $a$ and $b$ such that the fraction $\frac{a}{b}$ is written in the decimal system as $b.a.$
2012 Canadian Mathematical Olympiad Qualification Repechage, 3
We say that $(a,b,c)$ form a [i]fantastic triplet[/i] if $a,b,c$ are positive integers, $a,b,c$ form a geometric sequence, and $a,b+1,c$ form an arithmetic sequence. For example, $(2,4,8)$ and $(8,12,18)$ are fantastic triplets. Prove that there exist infinitely many fantastic triplets.
2004 Iran Team Selection Test, 6
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
2014 Dutch IMO TST, 1
Determine all pairs $(a,b)$ of positive integers satisfying
\[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]
2014 Greece Team Selection Test, 1
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.
2008 Irish Math Olympiad, 1
Find, with proof, all triples of integers $ (a,b,c)$ such that $ a, b$ and $ c$ are the lengths of the sides of a right angled triangle whose area is $ a \plus{} b \plus{} c$
2004 Iran Team Selection Test, 6
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
2024 Baltic Way, 17
Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?
2005 Iran MO (3rd Round), 3
$p(x)$ is an irreducible polynomial in $\mathbb Q[x]$ that $\mbox{deg}\ p$ is odd. $q(x),r(x)$ are polynomials with rational coefficients that $p(x)|q(x)^2+q(x).r(x)+r(x)^2$. Prove that \[p(x)^2|q(x)^2+q(x).r(x)+r(x)^2\]