IMO 2007

Real numbers $a_{1}$ , $a_{2}$ , $\ldots$ , $a_{n}$ are given. For each $i$ , $(1 \leq i \leq n )$ , define $$d_{i} = \max \{ a_{j}\mid 1 \leq j \leq i \} - \min \{ a_{j}\mid i \leq j \leq n \}$$ and let $d = \max \{d_{i}\mid 1 \leq i \leq n \}$ .

(a) Prove that, for any real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ , $$\max \{ |x_{i} - a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. (*)$$ (b) Show that there are real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).

Author: Michael Albert, New Zealand

Consider five points $A$ , $B$ , $C$ , $D$ and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$ . Suppose that $\ell$ intersects the interior of the segment $DC$ at $F$ and intersects line $BC$ at $G$ . Suppose also that $EF = EG = EC$ . Prove that $\ell$ is the bisector of angle $DAB$ .

Author: Charles Leytem, Luxembourg

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitiors is a clique.) The number of members of a clique is called its size.

Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.

Author: Vasily Astakhov, Russia

In triangle $ABC$ the bisector of angle $BCA$ intersects the circumcircle again at $R$ , the perpendicular bisector of $BC$ at $P$ , and the perpendicular bisector of $AC$ at $Q$ . The midpoint of $BC$ is $K$ and the midpoint of $AC$ is $L$ . Prove that the triangles $RPK$ and $RQL$ have the same area.

Author: Marek Pechal, Czech Republic

Let $a$ and $b$ be positive integers. Show that if $4ab - 1$ divides $(4a^{2} - 1)^{2}$ , then $a = b$ .

Author: Kevin Buzzard and Edward Crane, United Kingdom

Let $n$ be a positive integer. Consider $$S = \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x + y + z > 0 \right \}$$ as a set of $(n + 1)^{3} - 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $S$ but does not include $(0,0,0)$ .

Author: Gerhard Wöginger, Netherlands