Found problems: 396
2006 China Team Selection Test, 3
$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$.
Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.
2012 ELMO Shortlist, 6
Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$.
For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$.
[i]Linus Hamilton.[/i]
2009 AIME Problems, 13
The terms of the sequence $ (a_i)$ defined by $ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} 2009} {1 \plus{} a_{n \plus{} 1}}$ for $ n \ge 1$ are positive integers. Find the minimum possible value of $ a_1 \plus{} a_2$.
1999 Bundeswettbewerb Mathematik, 1
Exactly 1600 Coconuts are distributed on exactly 100 monkeys, where some monkeys also can have 0 coconuts.
Prove that, no matter how you distribute the coconuts, at least 4 monkeys will always have the same amount of coconuts.
(The original problem is written in German. So, I apologize when I've changed the original problem or something has become unclear while translating.)
2007 Romania Team Selection Test, 4
Let $S$ be the set of $n$-uples $\left( x_{1}, x_{2}, \ldots, x_{n}\right)$ such that $x_{i}\in \{ 0, 1 \}$ for all $i \in \overline{1,n}$, where $n \geq 3$. Let $M(n)$ be the smallest integer with the property that any subset of $S$ with at least $M(n)$ elements contains at least three $n$-uples \[\left( x_{1}, \ldots, x_{n}\right), \, \left( y_{1}, \ldots, y_{n}\right), \, \left( z_{1}, \ldots, z_{n}\right) \] such that
\[\sum_{i=1}^{n}\left( x_{i}-y_{i}\right)^{2}= \sum_{i=1}^{n}\left( y_{i}-z_{i}\right)^{2}= \sum_{i=1}^{n}\left( z_{i}-x_{i}\right)^{2}. \]
(a) Prove that $M(n) \leq \left\lfloor \frac{2^{n+1}}{n}\right\rfloor+1$.
(b) Compute $M(3)$ and $M(4)$.
PEN P Problems, 6
Show that every integer greater than $1$ can be written as a sum of two square-free integers.
2012 ELMO Shortlist, 4
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
2012 Argentina Cono Sur TST, 1
Sofía colours $46$ cells of a $9 \times 9$ board red. If Pedro can find a $2 \times 2$ square from the board that has $3$ or more red cells, he wins; otherwise, Sofía wins. Determine the player with the winning strategy.
2010 All-Russian Olympiad, 2
Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out.
1995 Korea National Olympiad, Problem 1
For any positive integer $m$,show that there exist integers $a,b$ satisfying
$\left | a \right |\leq m$, $ \left | b \right |\leq m$, $0< a+b\sqrt{2}\leq \frac{1+\sqrt{2}}{m+2}$
2007 All-Russian Olympiad Regional Round, 9.8
A set contains $ 372$ integers from $ 1,2,...,1200$ . For every element $ a\in S$, the numbers $ a\plus{}4,a\plus{}5,a\plus{}9$ don't belong to $ S$. Prove that $ 600\in S$.
2020 Cono Sur Olympiad, 2
Given $2021$ distinct positive integers non divisible by $2^{1010}$, show that it's always possible to choose $3$ of them $a$, $b$ and $c$, such that $|b^2-4ac|$ is not a perfect square.
2004 Romania National Olympiad, 4
In the interior of a cube of side $6$ there are $1001$ unit cubes with the faces parallel to the faces of the given cube. Prove that there are $2$ unit cubes with the property that the center of one of them lies in the interior or on one of the faces of the other cube.
[i]Dinu Serbanescu[/i]
2017 India Regional Mathematical Olympiad, 4
Consider \(n^2\) unit squares in the \(xy\) plane centered at point \((i,j)\) with integer coordinates, \(1 \leq i \leq n\), \(1 \leq j \leq n\). It is required to colour each unit square in such a way that whenever \(1 \leq i < j \leq n\) and \(1 \leq k < l \leq n\), the three squares with centres at \((i,k),(j,k),(j,l)\) have distinct colours. What is the least possible number of colours needed?
2016 Bundeswettbewerb Mathematik, 4
There are $33$ children in a given class. Each child writes a number on the blackboard, which indicates how many other children possess the same forename as oneself. Afterwards, each child does the same thing with their surname. After they've finished, each of the numbers $0,1,2,\dots,10$ appear at least once on the blackboard.
Prove that there are at least two children in this class that have the same forename and surname.
1996 Flanders Math Olympiad, 3
Consider the points $1,\frac12,\frac13,...$ on the real axis. Find the smallest value $k \in \mathbb{N}_0$ for which all points above can be covered with 5 [b]closed[/b] intervals of length $\frac1k$.
PEN Q Problems, 2
Prove that there is no nonconstant polynomial $f(x)$ with integral coefficients such that $f(n)$ is prime for all $n \in \mathbb{N}$.
2010 China Team Selection Test, 3
Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers
$a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.
2011 Preliminary Round - Switzerland, 3
On a blackboard, there are $11$ positive integers. Show that one can choose some (maybe all) of these numbers and place "$+$" and "$-$" in between such that the result is divisible by $2011$.
2003 Manhattan Mathematical Olympiad, 2
A tennis net is made of strings tied up together which make a grid consisting of small congruent squares as shown below.
[asy]
size(500);
xaxis(-50,50);
yaxis(-5,5);
add(shift(-50,-5)*grid(100,10));[/asy]
The size of the net is $100\times 10$ small squares. What is the maximal number of edges of small squares which can be cut without breaking the net into two pieces? (If an edge is cut, the cut is made in the middle, not at the ends.)
1995 Baltic Way, 20
All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$.
[i]Bogdan Enescu[/i]
2006 Iran MO (3rd Round), 4
The image shown below is a cross with length 2. If length of a cross of length $k$ it is called a $k$-cross. (Each $k$-cross ahs $6k+1$ squares.)
[img]http://aycu08.webshots.com/image/4127/2003057947601864020_th.jpg[/img]
a) Prove that space can be tiled with $1$-crosses.
b) Prove that space can be tiled with $2$-crosses.
c) Prove that for $k\geq5$ space can not be tiled with $k$-crosses.
2011 China Western Mathematical Olympiad, 2
Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition:
For any three elements in $M$, there exist two of them $a$ and $b$ such that $a|b$ or $b|a$.
Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$
2008 Mongolia Team Selection Test, 1
How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?
2011 Iran Team Selection Test, 12
Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.