IMO 1978 — Official contest problems

Let $n\gt m\ge1$ be natural numbers such that the groups of the last three digits in the decimal representation of $1978^m,1978^n$ coincide. Find the ordered pair $(m,n)$ of such $m,n$ for which $m+n$ is minimal.

Given any point $P$ in the interior of a sphere with radius $R$, three mutually perpendicular segments $PA,PB,PC$ are drawn terminating on the sphere and having one common vertex in $P$. Consider the rectangular parallelepiped of which $PA,PB,PC$ are coterminal edges. Find the locus of the point $Q$ that is diagonally opposite $P$ in the parallelepiped when $P$ and the sphere are fixed.

Let $\{f(n)\}$ be a strictly increasing sequence of positive integers: $0\lt f(1)\lt f(2)\lt f(3)\lt \cdots$. Of the positive integers not belonging to the sequence, the $n$th in order of magnitude is $f(f(n))+1$. Determine $f(240)$.

In a triangle $ABC$ we have $AB=AC$. A circle is tangent internally to the circumcircle of $ABC$ and also to the sides $AB,AC$, at $P,Q$ respectively. Prove that the midpoint of $PQ$ is the center of the incircle of $ABC$.

Let $\varphi:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ be injective. Prove that for all $n$,
$$\sum_{k=1}^n\frac{\varphi(k)}{k^2}\ge\sum_{k=1}^n\frac1k.$$

An international society has its members in $6$ different countries. The list of members contains $1978$ names, numbered $1,2,\ldots,1978$. Prove that there is at least one member whose number is the sum of the numbers of two, not necessarily distinct, of his compatriots.