International Mathematical Olympiad · 1978

Problems

Problem 1Let $m$ and $n$ be positive integers with $1 \le m < n$ . 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.Solutions: 1
Problem 2We consider a fixed point $P$ in the interior of a fixed sphere. We construct three segments $PA, PB, PC$ , perpendicular two by two, with the vertices $A, B, C$ on the sphere. We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges. Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.Solutions: 1
Problem 3Let $0<f(1)<f(2)<f(3)<\ldots$ be a sequence with all its terms positive integers. The $n$ -th positive integer which doesn't belong to the sequence is $f(f(n))+1$ . Find $f(240)$ .Solutions: 1
Problem 4In a triangle $ABC$ we have $AB=AC$ . A circle internally tangent to the circumcircle is also tangent to $AB,AC$ at $P,Q$ . Prove that the midpoint of $PQ$ is the incenter of triangle ABC.Solutions: 1
Problem 5Let $f$ be an injective function from $\{1,2,3,\ldots\}$ into itself. Prove that for any $n$ we have $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}$ .Solutions: 1
Problem 6An international society has its members from six 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 members from his own country, or twice as large as the number of one member from his own country.Solutions: 1