Found problems: 1766
2006 Iran Team Selection Test, 4
Let $n$ be a fixed natural number.
Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have
\[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]
2007 Greece Junior Math Olympiad, 2
If $n$ is is an integer such that $4n+3$ is divisible by $11,$ find the form of $n$ and the remainder of $n^{4}$ upon division by $11$.
1979 IMO Longlists, 34
Notice that in the fraction $\frac{16}{64}$ we can perform a simplification as $\cancel{\frac{16}{64}}=\frac 14$ obtaining a correct equality. Find all fractions whose numerators and denominators are two-digit positive integers for which such a simplification is correct.
2009 China Team Selection Test, 3
Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! \plus{} 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$
2012 Federal Competition For Advanced Students, Part 1, 1
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying the following property: For each pair of integers $m$ and $n$ (not necessarily distinct), $\mathrm{gcd}(m, n)$ divides $f(m) + f(n)$.
Note: If $n\in\mathbb{Z}$, $\mathrm{gcd}(m, n)=\mathrm{gcd}(|m|, |n|)$ and $\mathrm{gcd}(n, 0)=n$.
1991 Turkey Team Selection Test, 2
Show that the equation $a^2+b^2+c^2+d^2=a^2\cdot b^2\cdot c^2\cdot d^2$ has no solution in positive integers.
2006 Baltic Way, 17
Determine all positive integers $n$ such that $3^{n}+1$ is divisible by $n^{2}$.
2006 JBMO ShortLists, 8
Prove that there do not exist natural numbers $ n\ge 10$ having all digits different from zero, and such that all numbers which are obtained by permutations of its digits are perfect squares.
2008 IMS, 6
Let $ a_0,a_1,\dots,a_{n \plus{} 1}$ be natural numbers such that $ a_0 \equal{} a_{n \plus{} 1} \equal{} 1$, $ a_i>1$ for all $ 1\leq i \leq n$, and for each $ 1\leq j\leq n$, $ a_i|a_{i \minus{} 1} \plus{} a_{i \plus{} 1}$. Prove that there exist one $ 2$ in the sequence.
2007 Indonesia TST, 3
Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.
2001 Hungary-Israel Binational, 1
Find positive integers $x, y, z$ such that $x > z > 1999 \cdot 2000 \cdot 2001 > y$ and $2000x^{2}+y^{2}= 2001z^{2}.$
1992 Baltic Way, 1
Let $p,q$ be two consecutive odd prime numbers. Prove that $p+q$ is a product of at least $3$ natural numbers greater than $1$ (not necessarily different).
1974 IMO Longlists, 35
If $p$ and $q$ are distinct prime numbers, then there are integers $x_0$ and $y_0$ such that $1 = px_0 + qy_0.$ Determine the maximum value of $b - a$, where $a$ and $b$ are positive integers with the following property:
If $a \leq t \leq b$, and $t$ is an integer, then there are integers $x$ and $y$ with $0 \leq x \leq q - 1$ and $0 \leq y \leq p - 1$ such that $t = px + qy.$
2009 India IMO Training Camp, 3
Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following:
$ a_1 \equal{} a \\
a_2 \equal{} b \\
a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2})$.
Prove that there exists a natural number $ N$ such that $ a_n \equal{} gcd(a,b) \forall n\ge N$.
2024 Czech-Polish-Slovak Junior Match, 2
How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?
2009 JBMO TST - Macedonia, 1
On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number.
2009 Indonesia TST, 1
Prove that for all odd $ n > 1$, we have $ 8n \plus{} 4|C^{4n}_{2n}$.
2002 Polish MO Finals, 3
$k$ is a positive integer. The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = k+1$, $a_{n+1} = a_n ^2 - ka_n + k$. Show that $a_m$ and $a_n$ are coprime (for $m \not = n$).
2001 Baltic Way, 20
From a sequence of integers $(a, b, c, d)$ each of the sequences
\[(c, d, a, b),\quad (b, a, d, c),\quad (a + nc, b + nd, c, d),\quad (a + nb, b, c + nd, d)\]
for arbitrary integer $n$ can be obtained by one step. Is it possible to obtain $(3, 4, 5, 7)$ from $(1, 2, 3, 4)$ through a sequence of such steps?
1991 Brazil National Olympiad, 4
Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's).
1994 Baltic Way, 8
Show that for any integer $a\ge 5$ there exist integers $b$ and $c$, $c\ge b\ge a$, such that $a,b,c$ are the lengths of the sides of a right-angled triangle.
2004 Junior Balkan Team Selection Tests - Romania, 4
Find all positive integers $n$ for which there exist distinct positive integers $a_1,a_2,\ldots,a_n$ such that
\[ \frac 1{a_1} + \frac 2{a_2} + \cdots + \frac n { a_n} = \frac { a_1 + a_2 + \cdots + a_n } n. \]
2011 Switzerland - Final Round, 7
For a given rational number $r$, find all integers $z$ such that \[2^z + 2 = r^2\mbox{.}\]
[i](Swiss Mathematical Olympiad 2011, Final round, problem 7)[/i]
1996 Iran MO (3rd Round), 4
Let $n$ be a positive integer and suppose that $\phi(n)=\frac{n}{k}$, where $k$ is the greatest perfect square such that $k \mid n$. Let $a_1,a_2,\ldots,a_n$ be $n$ positive integers such that $a_i=p_1^{a_1i} \cdot p_2^{a_2i} \cdots p_n^{a_ni}$, where $p_i$ are prime numbers and $a_{ji}$ are non-negative integers, $1 \leq i \leq n, 1 \leq j \leq n$. We know that $p_i\mid \phi(a_i)$, and if $p_i\mid \phi(a_j)$, then $p_j\mid \phi(a_i)$. Prove that there exist integers $k_1,k_2,\ldots,k_m$ with $1 \leq k_1 \leq k_2 \leq \cdots \leq k_m \leq n$ such that
\[\phi(a_{k_{1}} \cdot a_{k_{2}} \cdots a_{k_{m}})=p_1 \cdot p_2 \cdots p_n.\]
2008 Bulgaria National Olympiad, 2
Is it possible to find $2008$ infinite arithmetical progressions such that there exist finitely many positive integers not in any of these progressions, no two progressions intersect and each progression contains a prime number bigger than $2008$?