International Mathematical Olympiad · 1962

Problems

Problem 1Find the smallest natural number whose decimal representation ends in $6$ and for which moving this final digit to the front produces four times the original number.Solutions: 1
Problem 2Determine all real numbers $x$ which satisfy $\sqrt{\sqrt{3-x}-\sqrt{x+1}}>\dfrac12$ .Solutions: 1
Problem 3The cube $ABCDA'B'C'D'$ has upper face $ABCD$ and lower face $A'B'C'D'$ , with $A$ directly above $A'$ and so on. Point $X$ moves at constant speed around $ABCD$ , and point $Y$ moves at the same speed around $B'C'CB$ . They leave $A$ toward $B$ and $B'$ toward $C'$ simultaneously. Find the locus of the midpoint of $XY$ .Solutions: 1
Problem 4Find all real solutions of $\cos^2 x+\cos^2 2x+\cos^2 3x=1$ .Solutions: 1
Problem 5Given three distinct points $A,B,C$ on a circle $K$ , construct a point $D$ on $K$ such that a circle can be inscribed in the convex quadrilateral $ABCD$ .Solutions: 1
Problem 6Consider an isosceles triangle. Let $R$ be the radius of its circumcircle and $r$ the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circles is $\sqrt{R(R-2r)}$ .Solutions: 1
Problem 7The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA,SB,SC,BC,CA,AB$ , or to their extensions. (a) Prove that the tetrahedron $SABC$ is regular. (b) Prove conversely that for every regular tetrahedron five such spheres exist.Solutions: 1