Found problems: 1766
2016 Middle European Mathematical Olympiad, 8
For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers.
Prove that:
1. There does not exist a solution $(a, b, c)$ for $n = 2017$.
2. For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$.
3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$.
2012 Iran MO (3rd Round), 3
Prove that for each $n \in \mathbb N$ there exist natural numbers $a_1<a_2<...<a_n$ such that $\phi(a_1)>\phi(a_2)>...>\phi(a_n)$.
[i]Proposed by Amirhossein Gorzi[/i]
2009 JBMO TST - Macedonia, 2
Let $ a $ and $ b $ be integer numbers. Let $ a = a^{2}+b^{2}-8b-2ab+16$. Prove that $ a $ is a perfect square.
2007 Indonesia MO, 2
For every positive integer $ n$, $ b(n)$ denote the number of positive divisors of $ n$ and $ p(n)$ denote the sum of all positive divisors of $ n$. For example, $ b(14)\equal{}4$ and $ p(14)\equal{}24$. Let $ k$ be a positive integer greater than $ 1$.
(a) Prove that there are infinitely many positive integers $ n$ which satisfy $ b(n)\equal{}k^2\minus{}k\plus{}1$.
(b) Prove that there are finitely many positive integers $ n$ which satisfy $ p(n)\equal{}k^2\minus{}k\plus{}1$.
2010 Kazakhstan National Olympiad, 2
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.
2024 All-Russian Olympiad, 8
Let $n>2$ be a positive integer. Masha writes down $n$ natural numbers along a circle. Next, Taya performs the following operation: Between any two adjacent numbers $a$ and $b$, she writes a divisor of the number $a+b$ greater than $1$, then Taya erases the original numbers and obtains a new set of $n$ numbers along the circle. Can Taya always perform these operations in such a way that after some number of operations, all the numbers are equal?
[i]Proposed by T. Korotchenko[/i]
2011 Romania Team Selection Test, 2
Prove that the set $S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}$ contains arithmetic progressions of any finite length, but no infinite arithmetic progressions.
[i]Vasile Pop[/i]
1971 IMO Longlists, 23
Find all integer solutions of the equation
\[x^2+y^2=(x-y)^3.\]
2008 Tournament Of Towns, 4
Given three distinct positive integers such that one of them is the average of the two others. Can the product of these three integers be the perfect 2008th power of a positive integer?
2009 China National Olympiad, 2
Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\plus{}5^q.$
2005 Romania Team Selection Test, 1
Solve the equation $3^x=2^xy+1$ in positive integers.
2009 China Team Selection Test, 2
Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$
2013 Stars Of Mathematics, 1
Prove that for any integers $a,b$, the equation $2abx^4 - a^2x^2 - b^2 - 1 = 0$ has no integer roots.
[i](Dan Schwarz)[/i]
2010 Spain Mathematical Olympiad, 1
A [i]pucelana[/i] sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only?
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 $\omega (1)=0$.
2005 Taiwan TST Round 1, 3
Find all positive integer triples $(x,y,z)$ such that
$x<y<z$, $\gcd (x,y)=6$, $\gcd (y,z)=10$, $\gcd (x,z)=8$, and lcm$(x,y,z)=2400$.
Note that the problems of the TST are not arranged in difficulty (Problem 1 of day 1 was probably the most difficult!)
2003 Costa Rica - Final Round, 6
Say a number is [i]tico[/i] if the sum of it's digits is a multiple of $2003$.
$\text{(i)}$ Show that there exists a positive integer $N$ such that the first $2003$ multiples, $N,2N,3N,\ldots 2003N$ are all tico.
$\text{(ii)}$ Does there exist a positive integer $N$ such that all it's multiples are tico?
2006 Kyiv Mathematical Festival, 5
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 $e^2+f^2=\frac{c^2+d^2}{p}.$
2013 China Second Round Olympiad, 1
Let $n$ be a positive odd integer , $a_1,a_2,\cdots,a_n$ be any permutation of the positive integers $1,2,\cdots,n$ . Prove that :$(a_1-1)(a^2_2-2)(a^3_3-3)\cdots (a^n_n-n)$ is an even number.
1999 CentroAmerican, 5
Let $a$ be an odd positive integer greater than 17 such that $3a-2$ is a perfect square. Show that there exist distinct positive integers $b$ and $c$ such that $a+b,a+c,b+c$ and $a+b+c$ are four perfect squares.
2007 ISI B.Math Entrance Exam, 2
Let $a$ and $b$ be two non-zero rational numbers such that the equation $ax^2+by^2=0$ has a non-zero solution in rational numbers . Prove that for any rational number $t$ , there is a solution of the equation $ax^2+by^2=t$.
2012 Iran MO (3rd Round), 2
Prove that there exists infinitely many pairs of rational numbers $(\frac{p_1}{q},\frac{p_2}{q})$ with $p_1,p_2,q\in \mathbb N$ with the following condition:
\[|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.\]
[i]Proposed by Mohammad Gharakhani[/i]
2009 Baltic Way, 10
Let $d(k)$ denote the number of positive divisors of a positive integer $k$. Prove that there exist in finitely many positive integers $M$ that cannot be written as
\[M=\left(\frac{2\sqrt{n}}{d(n)}\right)^2\]
for any positive integer $n$.
2008 IberoAmerican Olympiad For University Students, 1
Let $n$ be a positive integer that is not divisible by either $2$ or $5$.
In the decimal expansion of $\frac{1}{n}= 0.a_1a_2a_3\cdots$ a finite number of digits after the decimal point are chosen arbitrarily to be deleted.
Clearly the decimal number obtained by this procedure is also rational, so it's equal to $\frac{a}{b}$ for some integers $a,b$. Prove that $b$ is divisible by $n$.
2000 Hungary-Israel Binational, 2
For a given integer $d$, let us define $S = \{m^{2}+dn^{2}| m, n \in\mathbb{Z}\}$. Suppose that $p, q$ are two elements of $S$ , where $p$ is prime and $p | q$. Prove that $r = q/p$ also belongs to $S$ .