Found problems: 1766
2007 Peru IMO TST, 3
Let $N$ be a natural number which can be expressed in the form $a^{2}+b^{2}+c^{2}$, where $a,b,c$ are integers divisible by 3.
Prove that $N$ can be expressed in the form $x^{2}+y^{2}+z^{2}$, where $x,y,z$ are integers and any of them are divisible by 3.
2013 China Team Selection Test, 1
Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]
2005 CentroAmerican, 2
Show that the equation $a^{2}b^{2}+b^{2}c^{2}+3b^{2}-c^{2}-a^{2}=2005$ has no integer solutions.
[i]Arnoldo Aguilar, El Salvador[/i]
2011 ITAMO, 2
A sequence of positive integers $a_1, a_2,\ldots, a_n$ is called [i]ladder[/i] of length $n$ if it consists of $n$ consecutive integers in ascending order.
(a) Prove that for every positive integer $n$ there exist two ladders of length $n$, with no elements in common,
$a_1, a_2,\ldots, a_n$ and $b_1, b_2,\ldots, b_n$, such that for all $i$ between $1$ and $n$, the greatest common divisor of $a_i$ and $b_i$ is equal to $1$.
(b) Prove that for every positive integer $n$ there exist two ladders of length $n$, with no elements in common,
$a_1, a_2,\ldots, a_n$ and $b_1, b_2,\ldots, b_n$, such that for all $i$ between $1$ and $n$, the greatest common divisor of $a_i$ and $b_i$ is greater than $1$.
1995 Italy TST, 1
Determine all triples $(x,y,z)$ of integers greater than $1$ with the property that $x$ divides $yz-1$, $y$ divides $zx-1$ and $z$ divides $xy-1$.
2015 Turkey MO (2nd round), 6
Find all positive integers $n$ such that for any positive integer $a$ relatively prime to $n$, $2n^2 \mid a^n - 1$.
2010 Contests, 2
Given a fixed integer $k>0,r=k+0.5$,define
$f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$
where $[x]$ denotes the smallest integer not less than $x$.
prove that there exists integer $m$ such that $f^m(r)$ is an integer.
2010 Romania Team Selection Test, 2
(a) Given a positive integer $k$, prove that there do not exist two distinct integers in the open interval $(k^2, (k + 1)^2)$ whose product is a perfect square.
(b) Given an integer $n > 2$, prove that there exist $n$ distinct integers in the open interval $(k^n, (k + 1)^n)$ whose product is the $n$-th power of an integer, for all but a finite number of positive integers $k$.
[i]AMM Magazine[/i]
2009 Middle European Mathematical Olympiad, 12
Find all non-negative integer solutions of the equation
\[ 2^x\plus{}2009\equal{}3^y5^z.\]
2008 Irish Math Olympiad, 3
Determine, with proof, all integers $ x$ for which $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square.
1986 Flanders Math Olympiad, 3
Let $\{a_k\}_{k\geq 0}$ be a sequence given by $a_0 = 0$, $a_{k+1}=3\cdot a_k+1$ for $k\in \mathbb{N}$.
Prove that $11 \mid a_{155}$
1994 Baltic Way, 7
Let $p>2$ be a prime number and
\[1+\frac{1}{2^3}+\frac{1}{3^3}+\ldots +\frac{1}{(p-1)^3}=\frac{m}{n}\]
where $m$ and $n$ are relatively prime. Show that $m$ is a multiple of $p$.
2008 Bosnia And Herzegovina - Regional Olympiad, 3
Find all positive integers $ a$ and $ b$ such that $ \frac{a^{4}\plus{}a^{3}\plus{}1}{a^{2}b^{2}\plus{}ab^{2}\plus{}1}$ is an integer.
2006 All-Russian Olympiad, 2
If an integer $a > 1$ is given such that $\left(a-1\right)^3+a^3+\left(a+1\right)^3$ is the cube of an integer, then show that $4\mid a$.
2014 Iran Team Selection Test, 2
is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that
$i) \exists n\in \mathbb{N}:f(n)\neq n$
$ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$
2011 Romanian Master of Mathematics, 4
Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$).
Prove the following two claims:
i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$;
ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$.
[i](Romania) Dan Schwarz[/i]
2001 JBMO ShortLists, 5
Let $x_k=\frac{k(k+1)}{2}$ for all integers $k\ge 1$. Prove that for any integer $n \ge 10$, between the numbers $A=x_1+x_2 + \ldots + x_{n-1}$ and $B=A+x_n$ there is at least one square.
2007 France Team Selection Test, 1
For a positive integer $a$, $a'$ is the integer obtained by the following method: the decimal writing of $a'$ is the inverse of the decimal writing of $a$ (the decimal writing of $a'$ can begin by zeros, but not the one of $a$); for instance if $a=2370$, $a'=0732$, that is $732$.
Let $a_{1}$ be a positive integer, and $(a_{n})_{n \geq 1}$ the sequence defined by $a_{1}$ and the following formula for $n \geq 1$:
\[a_{n+1}=a_{n}+a'_{n}. \]
Can $a_{7}$ be prime?
2010 Kazakhstan National Olympiad, 4
It is given that for some $n \in \mathbb{N}$ there exists a natural number $a$, such that $a^{n-1} \equiv 1 \pmod{n}$ and that for any prime divisor $p$ of $n-1$ we have $a^{\frac{n-1}{p}} \not \equiv 1 \pmod{n}$.
Prove that $n$ is a prime.
2003 India IMO Training Camp, 2
Find all triples $(a,b,c)$ of positive integers such that
(i) $a \leq b \leq c$;
(ii) $\text{gcd}(a,b,c)=1$; and
(iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$.
2010 Polish MO Finals, 2
Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$.
2012 China Team Selection Test, 3
Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]
2010 ITAMO, 1
In a mathematics test number of participants is $N < 40$. The passmark is fixed at $65$. The test results are
the following:
The average of all participants is $66$, that of the promoted $71$ and that of the repeaters $56$.
However, due to an error in the wording of a question, all scores are increased by $5$. At this point
the average of the promoted participants becomes $75$ and that of the non-promoted $59$.
(a) Find all possible values of $N$.
(b) Find all possible values of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$.
2010 China Western Mathematical Olympiad, 8
Determine all possible values of integer $k$ for which there exist positive integers $a$ and $b$ such that $\dfrac{b+1}{a} + \dfrac{a+1}{b} = k$.
2001 India IMO Training Camp, 2
Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that :
\[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]