Found problems: 1362
1997 Brazil Team Selection Test, Problem 3
Find all positive integers $x>1, y$ and primes $p,q$ such that $p^{x}=2^{y}+q^{x}$
1998 China Team Selection Test, 3
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.
2005 Postal Coaching, 26
Let $a_1,a_2,\ldots a_n$ be real numbers such that their sum is equal to zero. Find the value of \[ \sum_{j=1}^{n} \frac{1}{a_j (a_j +a _{j+1}) (a_j + a_{j+1} + a_{j+2}) \ldots (a_j + \ldots a_{j+n-2})}. \]
where the subscripts are taken modulo $n$ assuming none of the denominators is zero.
2002 Mexico National Olympiad, 3
Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?
2013 Romania Team Selection Test, 2
Let $n$ be an integer larger than $1$ and let $S$ be the set of $n$-element subsets of the set $\{1,2,\ldots,2n\}$. Determine
\[\max_{A\in S}\left (\min_{x,y\in A, x \neq y} [x,y]\right )\] where $[x,y]$ is the least common multiple of the integers $x$, $y$.
2013 Dutch IMO TST, 1
Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.
2007 China National Olympiad, 3
Find a number $n \geq 9$ such that for any $n$ numbers, not necessarily distinct, $a_1,a_2, \ldots , a_n$, there exists 9 numbers $a_{i_1}, a_{i_2}, \ldots , a_{i_9}, (1 \leq i_1 < i_2 < \ldots < i_9 \leq n)$ and $b_i \in {4,7}, i =1, 2, \ldots , 9$ such that $b_1a_{i_1} + b_2a_{i_2} + \ldots + b_9a_{i_9}$ is a multiple of 9.
2012 Poland - Second Round, 3
Denote by $S(k)$ the sum of the digits in the decimal representation of $k$. Prove that there are infinitely many $n\in \mathbb{Z_{+}}$ such that: ${S(2^{n}+n})<S(2^{n})$.
1994 Taiwan National Olympiad, 3
Let $a$ be a positive integer such that $5^{1994}-1\mid a$. Prove that the expression of $a$ in base $5$ contains at least $1994$ nonzero digits.
2023 India IMO Training Camp, 2
For a positive integer $k$, let $s(k)$ denote the sum of the digits of $k$. Show that there are infinitely many natural numbers $n$ such that $s(2^n) > s(2^{n+1})$.
2009 APMO, 4
Prove that for any positive integer $ k$, there exists an arithmetic sequence $ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3}, ... ,\frac{a_k}{b_k}$ of rational numbers, where $ a_i, b_i$ are relatively prime positive integers for each $ i \equal{} 1,2,...,k$ such that the positive integers $ a_1, b_1, a_2, b_2, ..., a_k, b_k$ are all distinct.
2010 Rioplatense Mathematical Olympiad, Level 3, 2
Find the minimum and maximum values of $ S=\frac{a}{b}+\frac{c}{d} $ where $a$, $b$, $c$, $d$ are positive integers satisfying $a + c = 20202$ and $b + d = 20200$.
2006 China Team Selection Test, 2
Given positive integers $m$, $a$, $b$, $(a,b)=1$. $A$ is a non-empty subset of the set of all positive integers, so that for every positive integer $n$ there is $an \in A$ and $bn \in A$. For all $A$ that satisfy the above condition, find the minimum of the value of $\left| A \cap \{ 1,2, \cdots,m \} \right|$
2003 China Western Mathematical Olympiad, 3
Let $ n$ be a given positive integer. Find the smallest positive integer $ u_n$ such that for any positive integer $ d$, in any $ u_n$ consecutive odd positive integers, the number of them that can be divided by $ d$ is not smaller than the number of odd integers among $ 1, 3, 5, \ldots, 2n \minus{} 1$ that can be divided by $ d$.
2009 Croatia Team Selection Test, 4
Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n \plus{} 1$, $ n \plus{} 2$, $ n \plus{} 3$.
2004 Postal Coaching, 20
Three numbers $N,n,r$ are such that the digits of $N,n,r$ taken together are formed by $1,2,3,4,5,6,7,8,9$ without repetition.
If $N = n^2 - r$, find all possible combinations of $N,n,r$.
1988 IMO Longlists, 60
Given integers $a_1, \ldots, a_{10},$ prove that there exist a non-zero sequence $\{x_1, \ldots, x_{10}\}$ such that all $x_i$ belong to $\{-1,0,1\}$ and the number $\sum^{10}_{i=1} x_i \cdot a_i$ is divisible by 1001.
1989 IMO Longlists, 82
Let $ A$ be a set of positive integers such that no positive integer greater than 1 divides all the elements of $ A.$ Prove that any sufficiently large positive integer can be written as a sum of elements of $ A.$ (Elements may occur several times in the sum.)
2014 IberoAmerican, 3
Given a set $X$ and a function $f: X \rightarrow X$, for each $x \in X$ we define $f^1(x)=f(x)$ and, for each $j \ge 1$, $f^{j+1}(x)=f(f^j(x))$. We say that $a \in X$ is a fixed point of $f$ if $f(a)=a$. For each $x \in \mathbb{R}$, let $\pi (x)$ be the quantity of positive primes lesser or equal to $x$.
Given an positive integer $n$, we say that $f: \{1,2, \dots, n\} \rightarrow \{1,2, \dots, n\}$ is [i]catracha[/i] if $f^{f(k)}(k)=k$, for every $k=1, 2, \dots n$. Prove that:
(a) If $f$ is catracha, $f$ has at least $\pi (n) -\pi (\sqrt{n}) +1$ fixed points.
(b) If $n \ge 36$, there exists a catracha function $f$ with exactly $ \pi (n) -\pi (\sqrt{n}) + 1$ fixed points.
2011 Danube Mathematical Competition, 2
Let S be a set of positive integers such that: min { lcm (x, y) : x, y ∈ S, $x \neq y$ } $\ge$ 2 + max S.
Prove that $\displaystyle\sum\limits_{x \in S} \frac{1}{x} \le \frac{3}{2} $.
1987 IMO Longlists, 32
Solve the equation $28^x = 19^y +87^z$, where $x, y, z$ are integers.
2001 India IMO Training Camp, 2
Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.
2008 Baltic Way, 10
For a positive integer $ n$, let $ S(n)$ denote the sum of its digits. Find the largest possible value of the expression $ \frac {S(n)}{S(16n)}$.
2000 Taiwan National Olympiad, 1
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.
1991 Vietnam National Olympiad, 2
Let $k>1$ be an odd integer. For every positive integer n, let $f(n)$ be the greatest positive integer for which $2^{f(n)}$ divides $k^n-1$. Find $f(n)$ in terms of $k$ and $n$.