IrMO 2004

Each of the players in a tennis tournament played one match against each of the others. If every player won at least one match, show that there is a group A; B; C of three players for which A beat B, B beat C and C beat A.

Prove that there are only two real numbers $x$ such that $$(x-1)(x-2)(x-3)(x-4)(x-5)(x-6) = 720$$

Let $a,b\ge 0$ . Prove that $$\sqrt{2}\left(\sqrt{a(a+b)^3}+b\sqrt{a^2+b^2}\right)\le 3(a^2+b^2)$$ with equality if and only if $a=b$ .

$A$ and $B$ are distinct points on a circle $T$ . $C$ is a point distinct from $B$ such that $|AB|=|AC|$ , and such that $BC$ is tangent to $T$ at $B$ . Suppose that the bisector of $\angle ABC$ meets $AC$ at a point $D$ inside $T$ . Show that $\angle ABC>72^\circ$ .

Suppose $p,q$ are distinct primes and $S$ is a subset of $\{1,2,\dots ,p-1\}$ . Let $N(S)$ denote the number of solutions to the equation $$\sum_{i=1}^{q}x_i\equiv 0\mod p$$ where $x_i\in S$ , $i=1,2,\dots ,q$ . Prove that $N(S)$ is a multiple of $q$ .