International Mathematical Olympiad  – IMO 1965  Problems

Problem 1

Determine all values $x$ in the interval $0\leq x\leq 2\pi$ which satisfy the inequality\[2\cos x \leq \left| \sqrt{1+\sin 2x} – \sqrt{1-\sin 2x } \right| \leq \sqrt{2}.\]

Problem 2

Consider the system of equations\[a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = 0\]\[a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = 0\]\[a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = 0\]with unknowns $x_1$, $x_2$, $x_3$. The coefficients satisfy the conditions:

(a) $a_{11}$, $a_{22}$, $a_{33}$ are positive numbers;

(b) the remaining coefficients are negative numbers;

(c) in each equation, the sum of the coefficients is positive.

Prove that the given system has only the solution $x_1 = x_2 = x_3 = 0$.

Problem 3

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\varepsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\varepsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

Problem 4

Find all sets of four real numbers $x_1$, $x_2$, $x_3$, $x_4$ such that the sum of any one and the product of the other three is equal to $2$.

Problem 5

Consider $\triangle OAB$ with acute angle $AOB$. Through a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over (a) the side $AB$, (b) the interior of $\triangle OAB$?

Problem 6

In a plane a set of $n$ points ($n\geq 3$) is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.