Found problems: 1362
2003 Regional Competition For Advanced Students, 2
Find all prime numbers $ p$ with $ 5^p\plus{}4p^4$ is the square of an integer.
1999 Irish Math Olympiad, 2
Show that there is a positive number in the Fibonacci sequence which is divisible by $ 1000$.
2013 Polish MO Finals, 1
Find all solutions of the following equation in integers $x,y: x^4+ y= x^3+ y^2$
2007 Moldova National Olympiad, 8.4
Solve in equation: $ x^2+y^2+z^2+w^2=3(x+y+z+w) $ where $ x,y,z,w $ are positive integers.
2007 Pre-Preparation Course Examination, 12
Find all subsets of $\mathbb N$ like $S$ such that
\[\forall m,n \in S \implies \dfrac{m+n}{\gcd(m,n)} \in S \]
1997 Federal Competition For Advanced Students, P2, 2
A positive integer $ K$ is given. Define the sequence $ (a_n)$ by $ a_1\equal{}1$ and $ a_n$ is the $ n$-th natural number greater than $ a_{n\minus{}1}$ which is congruent to $ n$ modulo $ K$.
$ (a)$ Find an explicit formula for $ a_n$.
$ (b)$ What is the result if $ K\equal{}2?$
2011 Greece National Olympiad, 1
Solve in integers the equation
\[{x^3}{y^2}\left( {2y - x} \right) = {x^2}{y^4} - 36\]
2002 China Team Selection Test, 3
For positive integers $a,b,c$ let $ \alpha, \beta, \gamma$ be pairwise distinct positive integers such that
\[ \begin{cases}{c} \displaystyle a &= \alpha + \beta + \gamma, \\
b &= \alpha \cdot \beta + \beta \cdot \gamma + \gamma \cdot \alpha, \\
c^2 &= \alpha\beta\gamma. \end{cases} \]
Also, let $ \lambda$ be a real number that satisfies the condition
\[\lambda^4 -2a\lambda^2 + 8c\lambda + a^2 - 4b = 0.\]
Prove that $\lambda$ is an integer if and only if $\alpha, \beta, \gamma$ are all perfect squares.
2003 China Team Selection Test, 2
Let $x<y$ be positive integers and $P=\frac{x^3-y}{1+xy}$. Find all integer values that $P$ can take.
2007 Mexico National Olympiad, 1
Find all integers $N$ with the following property: for $10$ but not $11$ consecutive positive integers, each one is a divisor of $N$.
1998 All-Russian Olympiad, 3
Let $S(x)$ denote the sum of the decimal digits of $x$. Do there exist natural numbers $a,b,c$ such that \[ S(a+b)<5, \quad S(b+c)<5, \quad S(c+a)<5, \quad S(a+b+c)> 50? \]
1997 Vietnam Team Selection Test, 3
Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions:
1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$;
2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.
2014 Romania Team Selection Test, 3
Determine all positive integers $n$ such that all positive integers less than $n$ and coprime to $n$ are powers of primes.
2005 Miklós Schweitzer, 2
Let $(a_{n})_{n \ge 1}$ be a sequence of integers satisfying the inequality \[ 0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1 \] for all $n \ge 2$. Prove that the sequence $(a_{n})$ is periodic.
Any Hints or Sols for this hard problem?? :help:
2013 Pan African, 1
A positive integer $n$ is such that $n(n+2013)$ is a perfect square.
a) Show that $n$ cannot be prime.
b) Find a value of $n$ such that $n(n+2013)$ is a perfect square.
2004 South africa National Olympiad, 1
Let $a=1111\dots1111$ and $b=1111\dots1111$ where $a$ has forty ones and $b$ has twelve ones. Determine the greatest common divisor of $a$ and $b$.
2002 Germany Team Selection Test, 3
Determine all $(x,y) \in \mathbb{N}^2$ which satisfy $x^{2y} + (x+1)^{2y} = (x+2)^{2y}.$
2001 Bulgaria National Olympiad, 3
Let $p$ be a prime number congruent to $3$ modulo $4$, and consider the equation $(p+2)x^{2}-(p+1)y^{2}+px+(p+2)y=1$.
Prove that this equation has infinitely many solutions in positive integers, and show that if $(x,y) = (x_{0}, y_{0})$ is a solution of the equation in positive integers, then $p | x_{0}$.
2010 India IMO Training Camp, 5
Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers.
(Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)
1988 China National Olympiad, 6
Let $n$ ($n\ge 3$) be a natural number. Denote by $f(n)$ the least natural number by which $n$ is not divisible (e.g. $f(12)=5$). If $f(n)\ge 3$, we may have $f(f(n))$ in the same way. Similarly, if $f(f(n))\ge 3$, we may have $f(f(f(n)))$, and so on. If $\underbrace{f(f(\dots f}_{k\text{ times}}(n)\dots ))=2$, we call $k$ the “[i]length[/i]” of $n$ (also we denote by $l_n$ the “[i]length[/i]” of $n$). For arbitrary natural number $n$ ($n\ge 3$), find $l_n$ with proof.
2011 Indonesia TST, 1
Find all real number $x$ which could be represented as
$x = \frac{a_0}{a_1a_2 . . . a_n} + \frac{a_1}{a_2a_3 . . . a_n} + \frac{a_2}{a_3a_4 . . . a_n} + . . . + \frac{a_{n-2}}{a_{n-1}a_n} + \frac{a_{n-1}}{a_n}$ , with $n, a_1, a_2, . . . . , a_n$ are positive integers and $1 = a_0 \leq a_1 < a_2 < . . . < a_n$
2004 China Team Selection Test, 3
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:
(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.
Find the largest possible value of $ |S|$.
2011 International Zhautykov Olympiad, 3
Let $\mathbb{N}$ denote the set of all positive integers. An ordered pair $(a;b)$ of numbers $a,b\in\mathbb{N}$ is called [i]interesting[/i], if for any $n\in\mathbb{N}$ there exists $k\in\mathbb{N}$ such that the number $a^k+b$ is divisible by $2^n$. Find all [i]interesting[/i] ordered pairs of numbers.
2002 Italy TST, 3
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that
$(\text{i})$ $x$ and $y$ are relatively prime;
$(\text{ii})$ $x$ divides $y^2+m;$
$(\text{iii})$ $y$ divides $x^2+m.$
2011 China Western Mathematical Olympiad, 1
Does there exist any odd integer $n \geq 3$ and $n$ distinct prime numbers $p_1 , p_2, \cdots p_n$ such that all $p_i + p_{i+1} (i = 1,2,\cdots , n$ and $p_{n+1} = p_{1})$ are perfect squares?