Find the largest value of n, such that there exists a polygon with n sides, 2 adjacent sides of length 1, and all his diagonals have an integer length.

For every natural number $ t$, $ f(t)$ is the probability that if a fair coin is tossed $ t$ times, the number of times we get heads is 2008 more than the number of tails. What is the value of $ t$ for which $ f(t)$ attains its maximum? (if there is more than one, describe all of them)

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. Author: Kei Irie, Japan

Prove that: $ \sum_{i=1}^{n^2} \lfloor \frac{i}{3} \rfloor= \frac{n^2(n^2-1)}{6}$ For all $ n \in N$.

The sequence $ a_n$ is defined as follows: $ a_0=1, a_1=1, a_{n+1}=\frac{1+a_{n}^2}{a_{n-1}}$. Prove that all the terms of the sequence are integers.

P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.