Found problems: 1766
2008 Argentina Iberoamerican TST, 3
Show that exists a sequence of $ 100$ terms such that:
1)Every term is a perfect square
2) every term is greater than the one before it ( it is strictly increasing)
3)Every two terms of the sequence are relative prime
4) The average between two consecutive terms is also a perfect square
Daniel
2007 All-Russian Olympiad Regional Round, 10.7
Given an integer $ n>6$. Consider those integers $ k\in (n(n\minus{}1),n^{2})$ which are coprime with $ n$. Prove that the greatest common divisor of the considered numbers is $ 1$.
2014 Baltic Way, 20
Consider a sequence of positive integers $a_1, a_2, a_3, . . .$ such that for $k \geq 2$ we have $a_{k+1} =\frac{a_k + a_{k-1}}{2015^i},$ where $2015^i$ is the maximal power of $2015$ that divides $a_k + a_{k-1}.$ Prove that if this sequence is periodic then its period is divisible by $3.$
2007 Iran Team Selection Test, 1
Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$, such that for each $i\neq j, (a_{i},a_{j})=1$, and for each $n$, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$?
[i]By Omid Hatami[/i]
2007 All-Russian Olympiad Regional Round, 8.3
Determine if there exist prime numbers $ p_{1},p_{2},...,p_{2007}$ such that $ p_{2}|p_{1}^{2}\minus{}1,p_{3}|p_{2}^{2}\minus{}1,...,p_{1}|p_{2007}^{2}\minus{}1$.
2009 Ukraine National Mathematical Olympiad, 2
Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$
2007 Iran MO (3rd Round), 5
A hyper-primitive root is a k-tuple $ (a_{1},a_{2},\dots,a_{k})$ and $ (m_{1},m_{2},\dots,m_{k})$ with the following property:
For each $ a\in\mathbb N$, that $ (a,m) \equal{} 1$, has a unique representation in the following form:
\[ a\equiv a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}\dots a_{k}^{\alpha_{k}}\pmod{m}\qquad 1\leq\alpha_{i}\leq m_{i}\]
Prove that for each $ m$ we have a hyper-primitive root.
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]
2000 JBMO ShortLists, 9
Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2$.
2024 Czech-Polish-Slovak Junior Match, 2
Among all triples $(a,b,c)$ of natural numbers satisfying
\[(a+14\sqrt{3})(b-14c\sqrt{3})=2024,\]
determine the one with the maximal value of $a$.
2014 District Olympiad, 3
Let $A=\{1,3,3^2,\ldots, 3^{2014}\}$. We obtain a partition of $A$ if $A$ is written as a disjoint union of nonempty subsets.
[list=a]
[*]Prove that there is no partition of $A$ such that the product of elements in each subset is a square.
[*]Prove that there exists a partition of $A$ such that the sum of elements in each subset is a square.[/list]
2006 JBMO ShortLists, 7
Determine all numbers $ \overline{abcd}$ such that $ \overline{abcd}\equal{}11(a\plus{}b\plus{}c\plus{}d)^2$.
2005 German National Olympiad, 3
Let s be a positive real.
Consider a two-dimensional Cartesian coordinate system. A [i]lattice point[/i] is defined as a point whose coordinates in this system are both integers. At each lattice point of our coordinate system, there is a lamp.
Initially, only the lamp in the origin of the Cartesian coordinate system is turned on; all other lamps are turned off. Each minute, we additionally turn on every lamp L for which there exists another lamp M such that
- the lamp M is already turned on,
and
- the distance between the lamps L and M equals s.
Prove that each lamp will be turned on after some time ...
[b](a)[/b] ... if s = 13. [This was the problem for class 11.]
[b](b)[/b] ... if s = 2005. [This was the problem for classes 12/13.]
[b](c)[/b] ... if s is an integer of the form $s=p_1p_2...p_k$ if $p_1$, $p_2$, ..., $p_k$ are different primes which are all $\equiv 1\mod 4$. [This is my extension of the problem, generalizing both parts [b](a)[/b] and [b](b)[/b].]
[b](d)[/b] ... if s is an integer whose prime factors are all $\equiv 1\mod 4$. [This is ZetaX's extension of the problem, and it is stronger than [b](c)[/b].]
Darij
2009 Spain Mathematical Olympiad, 4
Find all the integer pairs $ (x,y)$ such that:
\[ x^2\minus{}y^4\equal{}2009\]
2014 All-Russian Olympiad, 1
Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$. Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$.
[i]N. Agakhanov[/i]
2012 Tuymaada Olympiad, 4
Let $p=1601$. Prove that if
\[\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n},\]
where we only sum over terms with denominators not divisible by $p$ (and the fraction $\dfrac {m} {n}$ is in reduced terms) then $p \mid 2m+n$.
[i]Proposed by A. Golovanov[/i]
1987 Canada National Olympiad, 1
Find all solutions of $a^2 + b^2 = n!$ for positive integers $a$, $b$, $n$ with $a \le b$ and $n < 14$.
2014 China Team Selection Test, 4
Let $k$ be a fixed odd integer, $k>3$. Prove: There exist infinitely many positive integers $n$, such that there are two positive integers $d_1, d_2$ satisfying $d_1,d_2$ each dividing $\frac{n^2+1}{2}$, and $d_1+d_2=n+k$.
2014 Iran MO (3rd Round), 4
$2 \leq d$ is a natural number.
$B_{a,b}$={$a,a+b,a+2b,...,a+db$}
$A_{c,q}$={$cq^n \vert n \in\mathbb{N}$}
Prove that there are finite prime numbers like $p$ such exists $a,b,c,q$ from natural numbers :
$i$ ) $ p \nmid abcq $
$ ii$ ) $A_{c,q} \equiv B_{a,b} (mod p ) $
(15 points )
2010 Poland - Second Round, 3
Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).
2006 Taiwan National Olympiad, 1
Find all integer solutions $(x,y)$ to the equation $\displaystyle \frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$.
2008 International Zhautykov Olympiad, 1
For each positive integer $ n$,denote by $ S(n)$ the sum of all digits in decimal representation of $ n$.
Find all positive integers $ n$,such that $ n\equal{}2S(n)^3\plus{}8$.
2013 China Team Selection Test, 1
Let $n\ge 2$ be an integer. $a_1,a_2,\dotsc,a_n$ are arbitrarily chosen positive integers with $(a_1,a_2,\dotsc,a_n)=1$. Let $A=a_1+a_2+\dotsb+a_n$ and $(A,a_i)=d_i$. Let $(a_2,a_3,\dotsc,a_n)=D_1, (a_1,a_3,\dotsc,a_n)=D_2,\dotsc, (a_1,a_2,\dotsc,a_{n-1})=D_n$.
Find the minimum of $\prod\limits_{i=1}^n\dfrac{A-a_i}{d_iD_i}$
2005 Taiwan National Olympiad, 1
Let $A$ be the sum of the first $2k+1$ positive odd integers, and let $B$ be the sum of the first $2k+1$ positive even integers. Show that $A+B$ is a multiple of $4k+3$.
2006 Germany Team Selection Test, 3
Is the following statement true?
For each positive integer $n$, we can find eight nonnegative integers $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$ such that $n=\frac{2^a-2^b}{2^c-2^d}\cdot\frac{2^e-2^f}{2^g-2^h}$.