IMO 1978

Let $m$ and $n$ be positive integers such that $1 \le m < n$ . In their decimal representations, the last three digits of $1978^m$ are equal, respectively, to the last three digits of $1978^n$ . Find $m$ and $n$ such that $m + n$ has its least value.

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$ . Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.