Select two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a point strictly within the octagon? Express your answer as $a + b$, where the probability is $\tfrac{a}{b}$ and $a$ and $b$ are relatively prime positive integers.

Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.

Prove that \[ \log_ab\plus{}\log_bc\plus{}\log_ca\le \log_ba\plus{}\log_cb\plus{}\log_ac\] for all $ 1<a\le b\le c$.

In the plane with coordinates $x,y$, find an example of a convex set $M$ that contains infinitely many lattice points (i.e. points with integer coordinates), but at the same time only finitely many lattice points from $M$ lie on each line in that plane.

Al has three red marbles and four blue marbles. He draws two different marbles at the same time. What is the probability that one is red and the other is blue?

Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$ [i]Vlad Matei[/i]

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

Find all functions $f:\mathbb{N}\to \mathbb{N}$ so that for every natural numbers $m,n$ :$f(n)+2mn+f(m)$ is a perfect square.

A $ 4\times 4$ table is divided into $ 16$ white unit square cells. Two cells are called neighbors if they share a common side. A [i]move[/i] consists in choosing a cell and the colors of neighbors from white to black or from black to white. After exactly $ n$ moves all the $ 16$ cells were black. Find all possible values of $ n$.

Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$ [i]Proposed by Igor Voronovich, Belarus[/i]

Determine all $2\times2$ integer matrices $A$ having the following properties: $1.$ the entries of $A$ are (positive) prime numbers, $2.$ there exists a $2\times2$ integer matrix $B$ such that $A=B^2$ and the determinant of $B$ is the square of a prime number.

Let $a\geq 4$ be a positive integer. Determine the smallest value of $n\geq 5$, $n\neq a$, such that $a$ can be represented in the form$$a=\frac{x_1^2+x_2^2+\cdots + x_n^ 2}{x_1x_2\cdots x_n}$$for a suitable choice of the $n$ positive integers $x_1,x_2,\ldots ,x_n$.

With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]

Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$. (a) Prove that $$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$ (b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence: $$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$

The world famous infinite hotel with infinitely many floors (where the floors and the rooms on each floor are numbered with the positive integers) is full of guests: each room is occupied by exactly one guest. The manager of the hotel wants to carpet the corridor on each floor, and an infinite set of carpets of finite length (numbered with the positive integers) was obtained. Every guest marked an infinite number of carpets that they liked. Luckily, any two guests living on a different floor share only a finite number of carpets that they both like. Prove that the carpets can be distributed among the floors in a way that for every guest there are only finitely many carpets they like that are placed on floors different from the one where the guest is. [i]Proposed by András Imolay, Budapest[/i]

Let $a, b, c$ be naturals. The equation $ax^2-bx + c = 0$ has two roots at $[0, 1]$. Prove that $a\ge 5$ and $b\ge 5$.

Given that $ABC$ is a triangle where $AB < AC$. On the half-lines $BA$ and $CA$ we take points $F$ and $E$ respectively such that $BF = CE = BC$. Let $M,N$ and $H$ be the mid-points of the segments $BF,CE$ and $BC$ respectively and $K$ and $O$ be the circumcenters of the triangles $ABC$ and $MNH$ respectively. We assume that $OK$ cuts $BE$ and $HN$ at the points $A_1$ and $B_1$ respectively and that $C_1$ is the point of intersection of $HN$ and $FE$. If the parallel line from $A_1$ to $OC_1$ cuts the line $FE$ at $D$ and the perpendicular from $A_1$ to the line $DB_1$ cuts $FE$ at the point $M_1$, prove that $E$ is the orthocenter of the triangle $A_1OM_1$.

\[\int_{-1}^1 \sqrt{\frac{1+x}{1-x}}+\sqrt{\frac{1-x}{1+x}}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

Prove that for all $n \in \mathbb{Z^+}$ and for all positive real numbers satisfying $a_1a_2...a_n=1$ \[ \displaystyle\sum_{i=1}^{n} \frac{a_i}{\sqrt{{a_i}^4+3}} \leq \frac{1}{2}\displaystyle\sum_{i=1}^{n} \frac{1}{a_i} \]

Let $S$ be the set of positive integers $n$ such that \[3\cdot \varphi (n)=n,\] where $\varphi (n)$ is the number of positive integers $k\leq n$ such that $\gcd (k, n)=1$. Find \[\sum_{n\in S} \, \frac{1}{n}.\] [i]Proposed by Nathan Ramesh

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

Find all functions $f$ from the reals to the reals such that \[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\] for all real $x,y$.

A sequence $(a_n)$ is defined as follows: $a_1 = p$ is a prime number with exactly $300$ nonzero digits, and for each $n \geq 1, a_{n+1}$ is the decimal period of $1/a_n$ multiplies by $2$. Determine $a_{2003}.$

Let $a,b,c,d$ be real numbers such that $a>b>c>d\geq 0$ and $a + d = b + c$. Show that \[ x^a + x^d \geq x^b + x^c \] for $x>0$.

An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $\, AB \,$ intersects altitude $\, CC' \,$ and its extension at points $\, M \,$ and $\, N \,$, and the circle with diameter $\, AC \,$ intersects altitude $\, BB' \,$ and its extensions at $\, P \,$ and $\, Q \,$. Prove that the points $\, M, N, P, Q \,$ lie on a common circle.