From a square of side length $1$, four identical triangles are removed, one at each corner, leaving a regular octagon. What is the area of the octagon?

Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

Find all triples of positive integers $(x, y, z)$ with $$\frac{xy}{z}+ \frac{yz}{x}+\frac{zx}{y}= 3$$

A pair of numbers are twin primes if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

Let $ABCD$ be a quadrilateral in which every angle is smaller than $180^o$. If the bisectors of angles $\angle DAB$ and $\angle DCB$ are parallel, prove that $\angle ADC = \angle ABC$

The vertices of a regular $2012$-gon are labelled with the numbers $1$ through $2012$ in some order. Call a vertex a peak if its label is larger than the label of its two neighbours, and a valley if its label is smaller than the label of its two neighbours. Show that the total number of peaks is equal to the total number of valleys.

Find all real numbers $x$ such that $$x^3 = \{(x + 1)^3\}$$where $\{y\}$ denotes the fractional part of $y$, i.e. the difference between $y$ and the largest integer less than or equal to $y$.

Let $ABCD$ be a trapezoid, with $AB \parallel CD$ (the vertices are listed in cyclic order). The diagonals of this trapezoid are perpendicular to one another and intersect at $O$. The base angles $\angle DAB$ and $\angle CBA$ are both acute. A point $M$ on the line sgement $OA$ is such that $\angle BMD = 90^o$, and a point $N$ on the line segment $OB$ is such that $\angle ANC = 90^o$. Prove that triangles $OMN$ and $OBA$ are similar.

Two courier companies offer services in the country of Old Aland. For any two towns, at least one of the companies offers a direct link in both directions between them. Additionally, each company is willing to chain together deliveries (so if they offer a direct link from $A$ to $B$, and $B$ to $C$, and $C$ to $D$ for instance, they will deliver from $A$ to $D$.) Show that at least one of the two companies must be able to deliver packages between any two towns in Old Aland.

Let $p(x)$ be a polynomial with integer coefficients, and let $a, b$ and $c$ be three distinct integers. Show that it is not possible to have $p(a) = b$, $p(b) = c$, and $p(c) = a$.

Chris and Michael play a game on a $5 \times 5$ board, initially containing some black and white counters as shown below: Chris begins by removing any black counter, and sliding a white counter from an adjacent square onto the empty square. From that point on, the players take turns. Michael slides a black counter onto an adjacent empty square, and Chris does the same with white counters (no more counters are removed). If a player has no legal move, then he loses. (a) Show that, even if Chris and Michael play cooperatively, the game will come to an end. (b) Which player has a winning strategy?

Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.