International Mathematical Olympiad · 2007

Problems

Problem 1Real numbers $a_1, a_2, \ldots, a_n$ 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 $x_1 \le x_2 \le \cdots \le x_n$ , $\max\{|x_i - a_i| : 1 \le i \le n\} \ge \dfrac{d}{2}$ . (b) Show that there exist real numbers $x_1 \le x_2 \le \cdots \le x_n$ such that equality holds in the inequality above.Solutions: 1
Problem 2Consider five points $A, B, C, D, E$ 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 $DAB$ .Solutions: 1
Problem 3In 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 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.Solutions: 1
Problem 4In 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.Solutions: 1
Problem 5Let $a$ and $b$ be positive integers. Show that if $4ab-1$ divides $(4a^2-1)^2$ , then $a=b$ .Solutions: 1
Problem 6Let $n$ be a positive integer. Consider $S=\{(x,y,z) : x,y,z\in\{0,1,\ldots,n\},\ x+y+z>0\}$ 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 $(0,0,0)$ .Solutions: 1