NO-ABEL 2006

Each square in an $n \times n$ table is painted black or white. The routes where two rows meet two columns, called a quartet if the remaining squares are the same color. (a) What is the largest possible number of black squares in a $4 \times 4$ table without quartets? (b) Is it possible to paint a $5 \times 5$ table so that it has no quartets?

a) Let $a$ and $b$ be two non-negative real numbers. Show that $a+b \ge \sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab}$ b) Let $a$ and $b$ be two real numbers in $[0, 3]$ . Show that $\sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab} \ge \frac{(a+b)^2}{2}$

(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection. (b) Show that there are infinitely many triples ( $k, n, m$ ) that are such that $k^2 + n^2 = 5m^2$ , where $k, n$ and $m$ are integers, and not all three have any in common prime factor.

Let $\gamma$ be the circumscribed circle about a right-angled triangle $ABC$ with right angle $C$ . Let $\delta$ be the circle tangent to the sides $AC$ and $BC$ and tangent to the circle $\gamma$ internally. (a) Find the radius $i$ of $\delta$ in terms of $a$ when $AC$ and $BC$ both have length $a$ . (b) Show that the radius $i$ is twice the radius of the inscribed circle of $ABC$ .