Found problems: 2008
2004 Canada National Olympiad, 4
Let $p$ be an odd prime. Prove that:
\[\displaystyle\sum_{k\equal{}1}^{p\minus{}1}k^{2p\minus{}1} \equiv \frac{p(p\plus{}1)}{2} \pmod{p^2}\]
2006 USAMO, 1
Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and
\[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \]
if and only if $s$ is not a divisor of $p-1$.
Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.
2003 China Team Selection Test, 2
Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.
2024 Nepal TST, P1
Let $a, b$ be positive integers. Prove that if $a^b + b^a \equiv 3 \pmod{4}$, then either $a+1$ or $b+1$ can't be written as the sum of two integer squares.
[i](Proposed by Orestis Lignos, Greece)[/i]
2005 Irish Math Olympiad, 1
Show that $ 2005^{2005}$ is a sum of two perfect squares, but not a sum of two perfect cubes.
2004 AMC 12/AHSME, 3
For how many ordered pairs of positive integers $ (x,y)$ is $ x \plus{} 2y \equal{} 100$?
$ \textbf{(A)}\ 33 \qquad \textbf{(B)}\ 49 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 99 \qquad \textbf{(E)}\ 100$
2009 Indonesia TST, 4
2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.
2011 China Team Selection Test, 2
Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.
1996 USAMO, 4
An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a [i]binary sequence of length [/i]$n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.
2014 EGMO, 4
Determine all positive integers $n\geq 2$ for which there exist integers $x_1,x_2,\ldots ,x_{n-1}$ satisfying the condition that if $0<i<n,0<j<n, i\neq j$ and $n$ divides $2i+j$, then $x_i<x_j$.
2006 Purple Comet Problems, 5
Find the sum of all positive integers less than $2006$ which are both multiples of six and one more than a multiple of seven.
1994 Romania TST for IMO, 1:
Find the smallest nomial of this sequence that $a_1=1993^{1994^{1995}}$ and
\[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases} \]
PEN D Problems, 11
During a break, $n$ children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of $n$ for which eventually, perhaps after many rounds, all children will have at least one candy each.
2012 All-Russian Olympiad, 1
Let $a_1,\ldots ,a_{10}$ be distinct positive integers, all at least $3$ and with sum $678$. Does there exist a positive integer $n$ such that the sum of the $20$ remainders of $n$ after division by $a_1,a_2,\ldots ,a_{10},2a_1,2a_2,\ldots ,2a_{10}$ is $2012$?
2000 Turkey Junior National Olympiad, 2
Find the least positive integer $n$ such that $15$ divides the product
\[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\]
, for every positive integers $a_1, a_2, \dots, a_{15}$.
2012 Romanian Master of Mathematics, 1
Given a finite number of boys and girls, a [i]sociable set of boys[/i] is a set of boys such that every girl knows at least one boy in that set; and a [i]sociable set of girls[/i] is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.)
[i](Poland) Marek Cygan[/i]
1998 Brazil Team Selection Test, Problem 1
Let N be a positive integer greater than 2. We number the vertices
of a regular 2n-gon clockwise with the numbers 1, 2, . . . ,N,−N,−N +
1, . . . ,−2,−1. Then we proceed to mark the vertices in the following way.
In the first step we mark the vertex 1. If ni is the vertex marked in the
i-th step, in the i+1-th step we mark the vertex that is |ni| vertices away
from vertex ni, counting clockwise if ni is positive and counter-clockwise
if ni is negative. This procedure is repeated till we reach a vertex that has
already been marked. Let $f(N)$ be the number of non-marked vertices.
(a) If $f(N) = 0$, prove that 2N + 1 is a prime number.
(b) Compute $f(1997)$.
1991 IMO Shortlist, 17
Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$
2012 ELMO Shortlist, 1
Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square.
[i]David Yang, Alex Zhu.[/i]
2004 Balkan MO, 2
Solve in prime numbers the equation $x^y - y^x = xy^2 - 19$.
2012 USA TSTST, 3
Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions:
(a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime.
(b) $n \le f(n) \le n+2012$ for all $n$.
Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.
2004 France Team Selection Test, 1
If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$.
Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?
2007 Singapore Team Selection Test, 3
Let $A,B,C$ be $3$ points on the plane with integral coordinates. Prove that there exists a point $P$ with integral coordinates distinct from $A,B$ and $C$ such that the interiors of the segments $PA,PB$ and $PC$ do not contain points with integral coordinates.
2007 Turkey MO (2nd round), 1
Let $k>1$ be an integer, $p=6k+1$ be a prime number and $m=2^{p}-1$ .
Prove that $\frac{2^{m-1}-1}{127m}$ is an integer.
1997 All-Russian Olympiad, 1
Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$.
[i]M. Sonkin[/i]