International Competitions 1979

Let $p$ and $q$ be natural numbers such that

$$\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{1318}+\frac{1}{1319}.$$

Prove that $p$ is divisible by 1979.

A 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 of the line-segments $A_iB_j$ for all $i,j=1,\ldots,5,$ is colored either red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of a different color. Show that all 10 sides of the top and bottom faces are the same color.

Two circles in a plane intersect. Let $A$ be one of the points of intersection. Starting simultaneously from $A$ two points move with constant speeds, each point travelling along its own circle in the same sense. The two points return to A simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that, at any time, the distances from $P$ to the moving points are equal.

Given a plane $\pi,$ a point $P$ in this plane and a point $Q$ not in $\pi,$ find all points $R$ in $\pi$ such that the ratio $(QP+PA)/QR$ is a maximum.

Find all real numbers $a$ for which there exist non-negative real numbers $x_1,x_2,x_3,x_4,x_5$ satisfying the relations

$$\sum_{k=1}^{5}kx_k=a,\quad\sum_{k=1}^{5}k^3x_k=a^2,\quad\sum_{k=1}^{5}k^5x_k=a^3.$$

Let $A$ and $E$ be opposite vertices of a regular octagon. A frog starts jumping at vertex $A$. From any vertex of the octagon except $E,$ it may jump to either of the two adjacent vertices. When it reaches vertex $E,$ the frog stops and stays there. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E.$ Prove that $a_{2n-1}=0,$

$$a_{2n}=\frac{1}{\sqrt{2}}(x^{n-1}-y^{n-1}),\quad n=1,2,3,\cdots,$$

where $x=2+\sqrt{2}$ and $y=2-\sqrt{2}.$

Note. A path of $n$ jumps is a sequence of vertices $(P_0,\ldots,P_n)$ such that

(i) $P_0=A, P_n=E;$

(ii) for every $i, 0\leq i\leq n-1, P_i$ is distinct from $E;$

(iii) for every $i, 0\leq i\leq n-1, P_i$ and $P_{i+1}$ are adjacent.