IMO 1978 1. The last three decimal digits of 1978 m are the same as the last three decimal digits of 1978 n .
P is a point inside a sphere.
Find the locus of Q for all possible sets of such rays from P. 3. A circle is tangent internally to the circumcircle of the triangle and also to AB, AC at P, Q respectively. Prove that the midpoint of PQ is the center of the incircle of the triangle. An international society has its members from six different countries. 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 the number of a member from his own country.