A positive integer is called [i]joli[/i] if it can be written as the arithmetic mean of two or more (not necessarily distinct) powers of two, and [i]superjoli[/i] if it can be written as the arithmetic mean of two or more distinct powers of two. For instance $7$ and $92$ are superjoli because $7=\frac{2^4+2^2+1}{3}$ and $92=\frac{2^8+2^4+2^2}{3}$. a) Prove that every positive integer is joli. b) Prove that no power of two is superjoli. c) Find the smallest positive integer different from a power of two that is not superjoli. [i]France Olympiad[/i]

Let $n > 3$ be a natural number, and let $x_1$, $x_2$, ..., $x_n$ be $n$ positive real numbers whose product is $1$. Prove the inequality \[ \frac {1}{1 + x_1 + x_1\cdot x_2} + \frac {1}{1 + x_2 + x_2\cdot x_3} + ... + \frac {1}{1 + x_n + x_n\cdot x_1} > 1. \]

A real function has been assigned to every cell of an $n \times n$ table. Prove that a function can be assigned to each row and each column of this table such that the function assigned to each cell is equivalent to the combination of functions assigned to the row and the column containing it.

[list=a] [*]Prove that for every real number $t$ such that $0 < t < \tfrac{1}{2}$ there exists a positive integer $n$ with the following property: for every set $S$ of $n$ positive integers there exist two different elements $x$ and $y$ of $S$, and a non-negative integer $m$ (i.e. $m \ge 0 $), such that \[ |x-my|\leq ty.\] [*]Determine whether for every real number $t$ such that $0 < t < \tfrac{1}{2} $ there exists an infinite set $S$ of positive integers such that \[|x-my| > ty\] for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m > 0$).

Let $A_1A_2\cdots A_{2002}$ be a regular 2002 sided polygon. Each vertex $A_i$ is associated with a positive integer $a_i$ such that the following condition is satisfied: If $j_1,j_2,\cdots, j_k$ are positive integers such that $k<500$ and $A_{j_1}, A_{j_2}, \cdots A_{j_k}$ is a regular $k$ sided polygon, then the values of $a_{j_1},A_{j_2}, \cdots A_{j_k}$ are all different. Find the smallest possible value of $a_1+a_2+\cdots a_{2002}$

Prove that if the natural numbers $ a $, $ b $, $ c $ satisfy the equation $$ a^2 + b^2 = c^2,$$ then: 1) at least one of the numbers $ a $ and $ b $ is divisible by $ 3 $, 2) at least one of the numbers $ a $ and $ b $ is divisible by $ 4 $, 3) at least one of the numbers $ a $, $ b $, $ c $ is divisible by $ 5 $.

$a_1, a_2, a_3, ...$ are positive reals such that $a_n^2 \ge a_1 + a_2 +... + a_{n-1}$. Show that for some $C > 0$ we have $a_n \ge C n$ for all $n$.

In a race among 5 snails, there is at most one tie, but that tie can involve any number of snails. For example, the result of the race might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second, and Bruna is fifth. How many different results of the race are possible? $ \textbf{(A) }180 \qquad \textbf{(B) }361 \qquad \textbf{(C) }420 \qquad \textbf{(D) }431 \qquad \textbf{(E) }720 \qquad $

a) A game master divides a group of $40$ players into four teams of ten. The players do not know what the teams are, however the master gives each player a card containing the names of two other players: one of them is a teammate and the other is not, but the master does not tell the player which is which. Can the master write the names on the cards in such a way that the players can determine the teams? (All of the players can work together to do so.) b) On the next occasion, the game master writes the names of $7$ teammates and $2$ opposing players on each card (possibly in a mixed up order). Now he wants to write the names in such a way that the players together cannot determine the four teams. Is it possible for him to achieve this? c) Can he write the names in such a way that the players together cannot determine the four teams, if now each card contains the names of $6$ teammates and $2$ opposing players (possibly in a mixed up order)?

Let $n$ be a natural number. There are $n$ boys and $n$ girls standing in a line, in any arbitrary order. A student $X$ will be eligible for receiving $m$ candies, if we can choose two students of opposite sex with $X$ standing on either side of $X$ in $m$ ways. Show that the total number of candies does not exceed $\frac 13n(n^2-1).$

A straight cone is given inside a rectangular parallelepiped $B$, with the apex at one of the vertices, say $T$ , of the parallelepiped, and the base touching the three faces opposite to $T .$ Its axis lies at the long diagonal through $T.$ If $V_1$ and $V_2$ are the volumes of the cone and the parallelepiped respectively, prove that \[V_1 \leq \frac{\sqrt 3 \pi V_2}{27}.\]

The value of $ \displaystyle{\frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}}}$ is $ \textbf{(A)}\ 3/4 \qquad \textbf{(B)}\ 4/5 \qquad \textbf{(C)}\ 5/6 \qquad \textbf{(D)}\ 6/7 \qquad \textbf{(E)}\ 6/5$

Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.

In a cyclic convex hexagon $ABCDEF$, $AB$ and $DC$ intersect at $G$, $AF$ and $DE$ intersect at $H$. Let $M, N$ be the circumcenters of $BCG$ and $EFH$, respectively. Prove that the $BE$, $CF$ and $MN$ are concurrent.

The diagram shows a magic square in which the sums of the numbers in any row, column or diagonal are equal. What is the value of $n$? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 11$

Show how to construct (by a ruler and a compass) a right-angled triangle, given its inradius and circumradius.

Find all positive integers $m,n$ such that the $m \times n$ grid can be tiled with figures formed by deleting one of the corners of a $2 \times 3$ grid. [i]usjl, ST[/i]

Let $x,y,z$ be positive real numbers, less than $\pi$, such that: $$\cos x + \cos y + \cos z = 0$$ $$\cos 2x + \cos 2 y + \cos 2z = 0$$ $$\cos 3x + \cos 3y + \cos 3z = 0$$ Find all the values that $\sin x + \sin y + \sin z$ can take.

Let $n>6$ be a perfect number, and let $n=p_1^{e_1}\cdot\cdot\cdot p_k^{e_k}$ be its prime factorisation with $1<p_1<\dots <p_k$. Prove that $e_1$ is an even number. A number $n$ is [i]perfect[/i] if $s(n)=2n$, where $s(n)$ is the sum of the divisors of $n$. (Proposed by Javier Rodrigo, Universidad Pontificia Comillas)

Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1.$$

Find all integers $n$ such that $2^n + n^2$ is a square of an integer. [i](Tomas Jurik )[/i]

A snail crawls over a table at a constant speed. Every $15$ minutes it turns by $90^o$, and in between these turns it crawls along a straight line. Prove that it can return to the starting point only in an integer number of hours.

Find all $f:\mathbb{R}\to\mathbb{R}$ surjective functions such that $$f(xf(y)+y^2)=f((x+y)^2)-xf(x) $$ for all real numbers $x,y$.

Dodecagon $QWARTZSPHINX$ has all side lengths equal to $2$, is not self-intersecting (in particular, the twelve vertices are all distinct), and moreover each interior angle is either $90^{\circ}$ or $270^{\circ}$. What are all possible values of the area of $\triangle SIX$?

Four men are each given a unique number from $1$ to $4$, and four women are each given a unique number from $1$ to $4$. How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are next to each other? Note that two configurations are considered to be the same if one can be rotated to obtain the other one.