International Mathematical Olympiad · 1961

Problems

Problem 1Solve $x+y+z=a,\quad x^2+y^2+z^2=b^2,\quad xy=z^2$ in real numbers. Find the necessary and sufficient condition on $a$ and $b$ for $x,y,z$ to be distinct positive numbers.Solutions: 1
Problem 2Let $a,b,c$ be the side lengths of a triangle whose area is $S$ . Prove that $a^2+b^2+c^2\ge4S\sqrt{3}$ . In what case does equality hold?Solutions: 1
Problem 3Solve the equation $\cos^n x-\sin^n x=1$ , where $n$ is a given positive integer.Solutions: 1
Problem 4Inside triangle $P_1P_2P_3$ a point $P$ is given. Let $Q_1,Q_2,Q_3$ be the intersections of $PP_1,PP_2,PP_3$ with the opposite sides. Prove that among $\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}$ there is one not larger than $2$ and one not smaller than $2$ .Solutions: 1
Problem 5Construct a triangle $ABC$ if $AC=b$ , $AB=c$ , and $\angle AMB=\omega$ with $\omega<90^\circ$ , where $M$ is the midpoint of $BC$ . Prove that the construction has a solution if and only if $b\tan(\omega/2)\le c<b$ . In what case does equality hold?Solutions: 1
Problem 6Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of it; the plane $ABC$ is not parallel to $\epsilon$ . In $\epsilon$ choose arbitrary points $A',B',C'$ . Let $L,M,N$ be the midpoints of $AA',BB',CC'$ , and let $G$ be the centroid of triangle $LMN$ . Find the locus of $G$ as $A',B',C'$ range independently over $\epsilon$ , excluding degenerate $LMN$ .Solutions: 1