IMO 1979

Problem 1. Let $p$ and $q$ be natural numbers such that

Prove that $p$ is divisible by $1979$.

Problem 2. A prism with pentagons $A_1{A}_2{A}_3{A}_4{A}_5$ and $B_1{B}_2{B}_3{B}_4{B}_5$ as top and bottom faces is given. Each side of the two pentagons and each of the line-segments $A_i{B}_j$ for all $i,j=1,...,5,$ 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.

Problem 3. 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.

Problem 4. 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.

Problem 5. 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

Problem 6. 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 $a_{2n-1}=0$,

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

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

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

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

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

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