International Mathematical Olympiad · 1967

Problems

Problem 1Let $ABCD$ be a parallelogram with $AB=a$ , $AD=1$ , and $\angle BAD=\alpha$ . If $\triangle ABD$ is acute, prove that the four circles of radius $1$ centered at $A,B,C,D$ cover the parallelogram if and only if $a\le\cos\alpha+\sqrt3\sin\alpha$ .Solutions: 1
Problem 2Prove that if one and only one edge of a tetrahedron is greater than $1$ , then its volume is at most $\frac18$ .Solutions: 1
Problem 3Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$ . Put $c_s=s(s+1)$ . Prove that $(c_{m+1}-c_k)(c_{m+2}-c_k)\cdots(c_{m+n}-c_k)$ is divisible by $c_1c_2\cdots c_n$ .Solutions: 1
Problem 4Let $A_0B_0C_0$ and $A_1B_1C_1$ be any two acute-angled triangles. Consider all triangles $ABC$ similar to $A_1B_1C_1$ (with corresponding vertices) and circumscribed about $A_0B_0C_0$ , where $A_0$ lies on $BC$ , $B_0$ on $CA$ , and $C_0$ on $AB$ . Determine and construct the triangle of maximum area.Solutions: 1
Problem 5Let $a_1,\ldots,a_8$ be real numbers, not all zero, and let $c_n=a_1^n+\cdots+a_8^n$ for $n=1,2,\ldots$ . Suppose infinitely many terms $c_n$ are zero. Determine all values of $n$ for which $c_n=0$ .Solutions: 1
Problem 6In a sports contest, $m$ medals were awarded on $n$ successive days ( $n>1$ ). On the first day, one medal and $\frac17$ of the remaining $m-1$ medals were awarded. On the second day, two medals and $\frac17$ of the then remaining medals were awarded, and so on. On the $n$ -th and last day, the remaining $n$ medals were awarded. Find $m$ and $n$ .Solutions: 1