IMO 2007

Problem 1. Real numbers $a_1,a_2,\dots ,a_n$ are given. For each $i$ ($1\le i\le n$) define

$d_i=\max \{a_j:1\le j\le i\}-\min \{a_j:i\le j\le n\}$

and let

$d=\max \{d_i:1\le i\le n\}.$

(a) Prove that, for any real numbers $x_1\le x_2\le \cdots \le x_n$,

$\max \{|x_i-a_i|:1\le i\le n\}\ge \frac{d}{2}.(\ast )$

(b) Show that there are real numbers $x_1\le x_2\le \cdots \le x_n$ such that equality holds in $(\ast )$.

Problem 2. 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 $DAB$.

Problem 3. 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 competitors 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 in 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.

Problem 4. 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.

Problem 5. Let $a$ and $b$ be positive integers. Show that if $4ab-1$ divides $(4a^2-1{)}^2$, then $a=b$.

Problem 6. Let $n$ be a positive integer. Consider

$S=\{(x,y,z):x,y,z\in \{0,1,\dots ,n\},x+y+z>0\}$

as a set of $(n+1{)}^3-1$ points in three-dimensional space. Determine the smallest possible number of planes, the union of which contains $S$ but does not include $(0,0,0)$.