International Mathematical Olympiad · 1979

Problems

Problem 1Let $p$ and $q$ be positive integers such that $\dfrac pq=1-\dfrac12+\dfrac13-\dfrac14+\cdots-\dfrac1{1318}+\dfrac1{1319}$ . Prove that $p$ is divisible by $1979$ .Solutions: 1
Problem 2A prism with pentagons $A_1A_2A_3A_4A_5$ and $B_1B_2B_3B_4B_5$ as top and bottom faces is given. Each side of the two pentagons and each segment $A_iB_j$ is colored red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of different colors. Prove that all 10 sides of the top and bottom faces have the same color.Solutions: 1
Problem 3Two circles in a plane intersect. Let $A$ be one of their points of intersection. Starting simultaneously from $A$ , two points move with constant speeds, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there exists a fixed point $P$ in the plane such that, at any time, the distances from $P$ to the moving points are equal.Solutions: 1
Problem 4Given a plane $\pi$ , a point $P$ in $\pi$ and a point $Q$ not in $\pi$ , find all points $R$ in $\pi$ such that the ratio $\dfrac{QP+PR}{QR}$ is a maximum.Solutions: 1
Problem 5Find all real numbers $a$ for which there exist non-negative real numbers $x_1,x_2,x_3,x_4,x_5$ satisfying $\sum_{k=1}^5kx_k=a$ , $\sum_{k=1}^5k^3x_k=a^2$ , and $\sum_{k=1}^5k^5x_k=a^3$ .Solutions: 1
Problem 6Let A and E be opposite vertices of a regular octagon. A frog starts at A and jumps to an adjacent vertex until it reaches E and stops. If a_n counts paths of exactly n jumps ending at E, prove $a_{2n-1}=0$ and $a_{2n}=\frac{(2+\sqrt{2})^{n-1}-(2-\sqrt{2})^{n-1}}{\sqrt{2}}$ for $n=1,2,3,\ldots$ .Solutions: 1