International Mathematical Olympiad · 1965

Problems

Problem 1Determine all values of $x$ in the interval $0 \le x \le 2\\pi$ which satisfy $2\\cos x \le |\\sqrt{1+\\sin 2x}-\\sqrt{1-\\sin 2x}| \le \\sqrt{2}$ .Solutions: 1
Problem 2Consider the system $a_{11}x_1+a_{12}x_2+a_{13}x_3=0$ , $a_{21}x_1+a_{22}x_2+a_{23}x_3=0$ , $a_{31}x_1+a_{32}x_2+a_{33}x_3=0$ . The diagonal coefficients $a_{11},a_{22},a_{33}$ are positive, the other coefficients are negative, and the sum of the coefficients in each equation is positive. Prove that the system has only the solution $x_1=x_2=x_3=0$ .Solutions: 1
Problem 3Given tetrahedron $ABCD$ , let opposite edges $AB$ and $CD$ have lengths $a$ and $b$ . Their skew-line distance is $d$ and their angle is $\theta$ . A plane parallel to both $AB$ and $CD$ divides the tetrahedron into two solids. The ratio of the plane's distances from $AB$ and $CD$ is $k$ . Compute the ratio of the volumes of the two solids.Solutions: 1
Problem 4Find all sets of four real numbers $x_1,x_2,x_3,x_4$ such that the sum of any one number and the product of the other three is equal to $2$ .Solutions: 1
Problem 5Let $\triangle OAB$ have acute angle $AOB$ . Through a point $M\ne O$ , draw perpendiculars to $OA$ and $OB$ , with feet $P$ and $Q$ . Let $H$ be the orthocenter of $\triangle OPQ$ . Find the locus of $H$ when $M$ ranges over (a) the side $AB$ ; (b) the interior of $\triangle OAB$ .Solutions: 1
Problem 6In the plane, let a set of $n$ points, $n\\ge3$ , be given, and join every pair by a segment. Let $d$ be the length of the longest segment. A diameter is any joining segment of length $d$ . Prove that the number of diameters is at most $n$ .Solutions: 1