Found problems: 1766
2011 Vietnam National Olympiad, 1
Define the sequence of integers $\langle a_n\rangle$ as;
\[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\]
Prove that $a_{2012}-2010$ is divisible by $2011.$
2005 CentroAmerican, 6
Let $n$ be a positive integer and $p$ a fixed prime. We have a deck of $n$ cards, numbered $1,\ 2,\ldots,\ n$ and $p$ boxes for put the cards on them. Determine all posible integers $n$ for which is possible to distribute the cards in the boxes in such a way the sum of the numbers of the cards in each box is the same.
2014 Contests, 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 $2011$.
2006 Kyiv Mathematical Festival, 4
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$
2012 Turkey MO (2nd round), 1
Find all polynomials with integer coefficients such that for all positive integers $n$ satisfies $P(n!)=|P(n)|!$
2008 Moldova Team Selection Test, 2
Let $ p$ be a prime number and $ k,n$ positive integers so that $ \gcd(p,n)\equal{}1$. Prove that $ \binom{n\cdot p^k}{p^k}$ and $ p$ are coprime.
2005 Italy TST, 3
The function $\psi : \mathbb{N}\rightarrow\mathbb{N}$ is defined by $\psi (n)=\sum_{k=1}^n\gcd (k,n)$.
$(a)$ Prove that $\psi (mn)=\psi (m)\psi (n)$ for every two coprime $m,n \in \mathbb{N}$.
$(b)$ Prove that for each $a\in\mathbb{N}$ the equation $\psi (x)=ax$ has a solution.
2005 Germany Team Selection Test, 3
A positive integer is called [i]nice[/i] if the sum of its digits in the number system with base $ 3$ is divisible by $ 3$.
Calculate the sum of the first $ 2005$ nice positive integers.
2010 China Girls Math Olympiad, 8
Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$, $a^2=m_2^2+n_2^2$, and $m_1-n_1=m_2-n_2.$
2010 All-Russian Olympiad Regional Round, 9.8
For every positive integer $n$, let $S_n$ be the sum of the first $n$ prime numbers:
$S_1 = 2, S_2 = 2 + 3 = 5, S_3 = 2 + 3 + 5 = 10$, etc. Can both $S_n$ and $S_{n+1}$
be perfect squares?
2000 Federal Competition For Advanced Students, Part 2, 2
Find all pairs of integers $(m, n)$ such that
\[ \left| (m^2 + 2000m+ 999999)- (3n^3 + 9n^2 + 27n) \right|= 1.\]
2006 Austrian-Polish Competition, 4
A positive integer $d$ is called [i]nice[/i] iff for all positive integers $x,y$ hold: $d$ divides $(x+y)^{5}-x^{5}-y^{5}$ iff $d$ divides $(x+y)^{7}-x^{7}-y^{7}$ .
a) Is 29 nice?
b) Is 2006 nice?
c) Prove that infinitely many nice numbers exist.
2010 ELMO Shortlist, 4
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
2010 Romania National Olympiad, 4
Let $a,b,c,d$ be positive integers, and let $p=a+b+c+d$. Prove that if $p$ is a prime, then $p$ is not a divisor of $ab-cd$.
[i]Marian Andronache[/i]
2011 Stars Of Mathematics, 2
Prove there do exist infinitely many positive integers $n$ such that if a prime $p$ divides $n(n+1)$ then $p^2$ also divides it (all primes dividing $n(n+1)$ bear exponent at least two).
Exhibit (at least) two values, one even and one odd, for such numbers $n>8$.
(Pál Erdös & Kurt Mahler)
2013 All-Russian Olympiad, 3
$100$ distinct natural numbers $a_1, a_2, a_3, \ldots, a_{100}$ are written on the board. Then, under each number $a_i$, someone wrote a number $b_i$, such that $b_i$ is the sum of $a_i$ and the greatest common factor of the other $99$ numbers. What is the least possible number of distinct natural numbers that can be among $b_1, b_2, b_3, \ldots, b_{100}$?
2005 Iran MO (3rd Round), 1
Find all $n,p,q\in \mathbb N$ that:\[2^n+n^2=3^p7^q\]
2008 Czech and Slovak Olympiad III A, 3
Find all pairs of integers $(a,b)$ such that $a^2+ab+1\mid b^2+ab+a+b-1$.
2010 Contests, 1
Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.
2006 Austrian-Polish Competition, 1
Let $M(n)=\{n,n+1,n+2,n+3,n+4,n+5\}$ be a set of 6 consecutive integers. Let's take all values of the form \[\frac{a}{b}+\frac{c}{d}+\frac{e}{f}\] with the set $\{a,b,c,d,e,f\}=M(n)$.
Let \[\frac{x}{u}+\frac{y}{v}+\frac{z}{w}=\frac{xvw+yuw+zuv}{uvw}\] be the greatest of all these values.
a) show: for all odd $n$ hold: $\gcd (xvw+yuw+zuv, uvw)=1$ iff $\gcd (x,u)=\gcd (y,v)=\gcd (z,w)=1$.
b) for which positive integers $n$ hold $\gcd (xvw+yuw+zuv, uvw)=1$?
2009 Italy TST, 3
Find all pairs of integers $(x,y)$ such that
\[ y^3=8x^6+2x^3y-y^2.\]
2004 IberoAmerican, 1
Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a\plus{}b$ and $ 201a\plus{}b$ are both perfect squares.
1997 Iran MO (3rd Round), 1
Suppose that $a, b, x$ are positive integers such that
\[x^{a+b}=a^bb\]
Prove that $a=x$ and $b=x^x$.
2004 Germany Team Selection Test, 1
Consider the real number axis (i. e. the $x$-axis of a Cartesian coordinate system). We mark the points $1$, $2$, ..., $2n$ on this axis. A flea starts at the point $1$. Now it jumps along the real number axis; it can jump only from a marked point to another marked point, and it doesn't visit any point twice. After the ($2n-1$)-th jump, it arrives at a point from where it cannot jump any more after this rule, since all other points are already visited. Hence, with its $2n$-th jump, the flea breaks this rule and gets back to the point $1$. Assume that the sum of the (non-directed) lengths of the first $2n-1$ jumps of the flea was $n\left(2n-1\right)$. Show that the length of the last ($2n$-th) jump is $n$.
2019 Turkey MO (2nd round), 6
Given an integer $n>2$ and an integer $a$, if there exists an integer $d$ such that $n\mid a^d-1$ and $n\nmid a^{d-1}+\cdots+1$, we say [i]$a$ is $n-$separating[/i]. Given any n>2, let the [i]defect of $n$[/i] be defined as the number of integers $a$ such that $0<a<n$, $(a,n)=1$, and $a$ is not [i] $n-$separating[/i]. Determine all integers $n>2$ whose defect is equal to the smallest possible value.