IMO 1965

Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality $$2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}.$$

Consider the sytem 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 ofthe coefficients is positive.

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

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 $\epsilon$ , parallel to lines $AB$ and $CD$ . The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$ . Compute the ratio of the volumes of the two solids obtained.

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.

Consider $\triangle OAB$ with acute angle $AOB$ . Thorugh 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 $\triangle OAB$ .

In a plane a set of $n\geq 3$ points 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 $n$ .