MEMO 2011

Let $ABCDE$ be a convex pentagon with all five sides equal in length. The diagonals $AD$ and $EC$ meet in $S$ with $\angle ASE = 60^\circ$ . Prove that $ABCDE$ has a pair of parallel sides.

Let $ABC$ be an acute triangle. Denote by $B_0$ and $C_0$ the feet of the altitudes from vertices $B$ and $C$ , respectively. Let $X$ be a point inside the triangle $ABC$ such that the line $BX$ is tangent to the circumcircle of the triangle $AXC_0$ and the line $CX$ is tangent to the circumcircle of the triangle $AXB_0$ . Show that the line $AX$ is perpendicular to $BC$ .

Let $A$ and $B$ be disjoint nonempty sets with $A \cup B = \{1, 2,3, \ldots, 10\}$ . Show that there exist elements $a \in A$ and $b \in B$ such that the number $a^3 + ab^2 + b^3$ is divisible by $11$ .

We call a positive integer $n$ amazing if there exist positive integers $a, b, c$ such that the equality $$n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)$$ holds. Prove that there exist $2011$ consecutive positive integers which are amazing.Note. By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$ .