Found problems: 1362
1995 Vietnam Team Selection Test, 3
Find all integers $ a$, $ b$, $ n$ greater than $ 1$ which satisfy
\[ \left(a^3 \plus{} b^3\right)^n \equal{} 4(ab)^{1995}
\]
2009 Hong Kong TST, 6
Show that the equation $ y^{37}\equiv x^3\plus{}11 \pmod p$ is solvable for every prime $ p$, where $ p\leq100$.
1976 IMO Longlists, 10
Show that the reciprocal of any number of the form $2(m^2+m+1)$, where $m$ is a positive integer, can be represented as a sum of consecutive terms in the sequence $(a_j)_{j=1}^{\infty}$
\[ a_j = \frac{1}{j(j + 1)(j + 2)}\]
2008 Brazil National Olympiad, 2
Prove that for all integers $ a > 1$ and $ b > 1$ there exists a function $ f$ from the positive integers to the positive integers such that $ f(a\cdot f(n)) \equal{} b\cdot n$ for all $ n$ positive integer.
2006 MOP Homework, 6
Find all integers $n$ for which there exists an equiangular $n$-gon whose side lengths are distinct rational numbers.
2010 Tuymaada Olympiad, 2
We have a number $n$ for which we can find 5 consecutive numbers, none of which is divisible by $n$, but their product is.
Show that we can find 4 consecutive numbers, none of which is divisible by $n$, but their product is.
1986 Iran MO (2nd round), 4
Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.
2009 German National Olympiad, 2
Find all positive interger $ n$ so that $ n^3\minus{}5n^2\plus{}9n\minus{}6$ is perfect square number.
2010 Contests, 4
Prove that for each positive integer n,the equation
$x^{2}+15y^{2}=4^{n}$
has at least $n$ integer solution $(x,y)$
2001 Bundeswettbewerb Mathematik, 2
For each $ n \in \mathbb{N}$ we have two numbers $ p_n, q_n$ with the following property: For exactly $ n$ distinct integer numbers $ x$ the number \[ x^2 \plus{} p_n \cdot x \plus{} q_n\] is the square of a natural number. (Note the definition of natural numbers includes the zero here.)
2013 Albania Team Selection Test, 5
Let $k$ be a natural number.Find all the couples of natural numbers $(n,m)$ such that :
$(2^k)!=2^n*m$
2000 All-Russian Olympiad, 2
Prove that one can partition the set of natural numbers into $100$ nonempty subsets such that among any three natural numbers $a$, $b$, $c$ satisfying $a+99b=c$, there are two that belong to the same subset.
1989 China Team Selection Test, 2
Let $v_0 = 0, v_1 = 1$ and $v_{n+1} = 8 \cdot v_n - v_{n-1},$ $n = 1,2, ...$. Prove that in the sequence $\{v_n\}$ there aren't terms of the form $3^{\alpha} \cdot 5^{\beta}$ with $\alpha, \beta \in \mathbb{N}.$
1978 IMO Longlists, 18
Given a natural number $n$, prove that the number $M(n)$ of points with integer coordinates inside the circle $(O(0, 0),\sqrt{n})$ satisfies
\[\pi n - 5\sqrt{n} + 1<M(n) < \pi n+ 4\sqrt{n} + 1\]
1999 All-Russian Olympiad, 2
Find all bounded sequences $(a_n)_{n=1}^\infty$ of natural numbers such that for all $n \ge 3$, \[ a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}. \]
2013 ELMO Problems, 3
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
2012 Indonesia TST, 4
Given a non-zero integer $y$ and a positive integer $n$. If $x_1, x_2, \ldots, x_n \in \mathbb{Z} - \{0, 1\}$ and $z \in \mathbb{Z}^+$ satisfy $(x_1x_2 \ldots x_n)^2y \le 2^{2(n+1)}$ and $x_1x_2 \ldots x_ny = z + 1$, prove that there is a prime among $x_1, x_2, \ldots, x_n, z$.
[color=blue]It appears that the problem statement is incorrect; suppose $y = 5, n = 2$, then $x_1 = x_2 = -1$ and $z = 4$. They all satisfy the problem's conditions, but none of $x_1, x_2, z$ is a prime. What should the problem be, or did I misinterpret the problem badly?[/color]
2010 Postal Coaching, 1
Does there exist an increasing sequence of positive integers $a_1 , a_2 ,\cdots$ with the following two properties?
(i) Every positive integer $n$ can be uniquely expressed in the form $n = a_j - a_i$ ,
(ii) $\frac{a_k}{k^3}$ is bounded.
2006 Estonia Math Open Senior Contests, 9
In the sequence $ (a_n)$ with general term $ a_n \equal{} n^3 \minus{} (2n \plus{} 1)^2$, does there exist a term that is divisible by 2006?
2010 Contests, 3
Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$,
\[a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots + a^{1!} + 1.\]
[i]Proposed by Milos Milosavljevic[/i]
2005 Balkan MO, 2
Find all primes $p$ such that $p^2-p+1$ is a perfect cube.
2014 Singapore MO Open, 5
Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1<k<n$ and $\gcd(k, n)=1$ is a prime.
1978 IMO Longlists, 27
Determine the sixth number after the decimal point in the number $(\sqrt{1978} +\lfloor\sqrt{1978}\rfloor)^{20}$
2002 Canada National Olympiad, 2
Call a positive integer $n$ [b]practical[/b] if every positive integer less than or equal to $n$ can be written as the sum of distinct divisors of $n$.
For example, the divisors of 6 are 1, 2, 3, and 6. Since
\[ \centerline{1={\bf 1}, ~~ 2={\bf 2}, ~~ 3={\bf 3}, ~~ 4={\bf 1}+{\bf 3}, ~~ 5={\bf 2}+ {\bf 3}, ~~ 6={\bf 6},} \]
we see that 6 is practical.
Prove that the product of two practical numbers is also practical.
1998 All-Russian Olympiad, 6
Are there $1998$ different positive integers, the product of any two being divisible by the square of their difference?