Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.

Find the smallest real number $p$ such that the inequality $\sqrt{1^2+1}+\sqrt{2^2+1}+...+\sqrt{n^2+1} \le \frac{1}{2}n(n+p)$ holds for all natural numbers $n$.

Let $M$ be a positive integer. At a party with 120 people, 30 wear red hats, 40 wear blue hats, and 50 wear green hats. Before the party begins, $M$ pairs of people are friends. (Friendship is mutual.) Suppose also that no two friends wear the same colored hat to the party. During the party, $X$ and $Y$ can become friends if and only if the following two conditions hold: [list] [*] There exists a person $Z$ such that $X$ and $Y$ are both friends with $Z$. (The friendship(s) between $Z,X$ and $Z,Y$ could have been formed during the party.) [*] $X$ and $Y$ are not wearing the same colored hat. [/list] Suppose the party lasts long enough so that all possible friendships are formed. Let $M_1$ be the largest value of $M$ such that regardless of which $M$ pairs of people are friends before the party, there will always be at least one pair of people $X$ and $Y$ with different colored hats who are not friends after the party. Let $M_2$ be the smallest value of $M$ such that regardless of which $M$ pairs of people are friends before the party, every pair of people $X$ and $Y$ with different colored hats are friends after the party. Find $M_1+M_2$. [hide="Clarifications"] [list] [*] The definition of $M_2$ should read, ``Let $M_2$ be the [i]smallest[/i] value of $M$ such that...''. An earlier version of the test read ``largest value of $M$''.[/list][/hide] [i]Victor Wang[/i]

Freddy writes down numbers $1, 2,\ldots ,n$ in some order. Then he makes a list of all pairs $(i, j)$ such that $1\le i<j\le n$ and the $i$-th number is bigger than the $j$-th number in his permutation. After that, Freddy repeats the following action while possible: choose a pair $(i, j)$ from the current list, interchange the $i$-th and the $j$-th number in the current permutation, and delete $(i, j)$ from the list. Prove that Freddy can choose pairs in such an order that, after the process finishes, the numbers in the permutation are in ascending order.

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

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.

Let $ S$ be the smallest subset of the integers with the property that $ 0\in S$ and for any $ x\in S$, we have $ 3x\in S$ and $ 3x \plus{} 1\in S$. Determine the number of non-negative integers in $ S$ less than $ 2008$.

Let $m$ and $n$ be integers greater than 1. Prove that $\left\lfloor \dfrac{mn}{6} \right\rfloor$ non-overlapping 2-by-3 rectangles can be placed in an $m$-by-$n$ rectangle. Note: $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.

Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k \minus{} n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either [i]on[/i] or [i]off[/i]. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off. Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off, but where none of the lamps $ n \plus{} 1$ through $ 2n$ is ever switched on. Determine $ \frac {N}{M}$. [i]Author: Bruno Le Floch and Ilia Smilga, France[/i]

Let $a$ be an odd natural number and $b$ be a positive integer. We define a sequence of reals $(u_n)$ as follows: $u_0=b$ and, for all $n\in\mathbb N_0$, $u_{n+1}$ is $\frac{u_n}2$ if $u_n$ is even and $a+u_n$ otherwise. (a) Prove that one can find an element of $u_n$ smaller than $a$. (b) Prove that the sequence is eventually periodic.

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. [b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \] [b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]

The set of positive nonzero real numbers are partitioned into three mutually disjoint non-empty subsets $(A\cup B\cup C)$. a) show that there exists a triangle of side-lengths $a,b,c$, such that $a\in A, b\in B, c\in C$. b) does it always happen that there exists a right triangle with the above property ?

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

We consider the sums of the form $\pm 1 \pm 4 \pm 9\pm ... \pm n^2$. Show that every integer can be represented in this form for some $n$. (For example, $3 = -1 + 4$ and $8 = 1-4-9+16+25-36-49+64$.)