International Mathematical Olympiad · 1970

Problems

Problem 1Let $M$ be a point on side $AB$ of triangle $ABC$ . Let $r_1,r_2,r$ be the inradii of triangles $AMC$ , $BMC$ , and $ABC$ , respectively. Let $q_1,q_2,q$ be the radii of the excircles of these triangles lying in angle $\angle ACB$ . Prove that $\frac{r_1}{q_1}\cdot\frac{r_2}{q_2}=\frac rq$ .Solutions: 2
Problem 2Let $a,b,n>1$ be integers and let $a,b$ be bases of two number systems. Using the same digits $x_n,\ldots,x_0$ , with $x_n\ne0$ and $x_{n-1}\ne0$ , define $A_n=x_nx_{n-1}\cdots x_0$ and $A_{n-1}=x_{n-1}\cdots x_0$ in base $a$ , and $B_n=x_nx_{n-1}\cdots x_0$ and $B_{n-1}=x_{n-1}\cdots x_0$ in base $b$ . Prove that $\frac{A_{n-1}}{A_n}<\frac{B_{n-1}}{B_n}$ if and only if $a>b$ .Solutions: 1
Problem 3Let $1=a_0\le a_1\le a_2\le\cdots$ be a sequence of real numbers. Define $b_n=\sum_{k=1}^n\left(1-\frac{a_{k-1}}{a_k}\right)\frac1{\sqrt{a_k}}$ . (a) Prove that $0\le b_n<2$ for every $n$ . (b) Given any $b$ with $0\le b<2$ , prove that there is such a sequence for which $b_n>b$ for infinitely many $n$ .Solutions: 1
Problem 4Find all positive integers $n$ such that $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two disjoint sets whose products are equal.Solutions: 2
Problem 5In tetrahedron $ABCD$ , $\angle BDC=90^\circ$ . The foot $H$ of the perpendicular from $D$ to plane $ABC$ is the intersection of the altitudes of $\triangle ABC$ . Prove that $(AB+BC+CA)^2\le6(AD^2+BD^2+CD^2)$ . For what tetrahedra does equality hold?Solutions: 1
Problem 6In a plane there are $100$ points, no three collinear. Consider all triangles whose vertices are among these points. Prove that no more than $70\%$ of these triangles are acute-angled.Solutions: 1