Question bank IMO 1965

(YUG) Find all real numbers $x\in[0,2\pi]$ such that
$$2\cos x\leq\left|\sqrt{1+\sin2x}-\sqrt{1-\sin2x}\right|\leq\sqrt2.$$

(POL) Consider the system of equations
$$\begin{cases}a_{11}x_1+a_{12}x_2+a_{13}x_3=0,\\a_{21}x_1+a_{22}x_2+a_{23}x_3=0,\\a_{31}x_1+a_{32}x_2+a_{33}x_3=0,\end{cases}$$
whose coefficients satisfy the following conditions:
(a) $a_{11},a_{22},a_{33}$ are positive real numbers;
(b) all other coefficients are negative;
(c) in each of the equations the sum of the coefficients is positive.
Prove that $x_1=x_2=x_3=0$ is the only solution to the system.

(CZS) A tetrahedron $ABCD$ is given. The lengths of the edges $AB$ and $CD$ are $a$ and $b$, respectively, the distance between the lines $AB$ and $CD$ is $d$, and the angle between them is equal to $\omega$. The tetrahedron is divided into two parts by the plane $\pi$ parallel to the lines $AB$ and $CD$. Calculate the ratio of the volumes of the parts if the ratio between the distances of the plane $\pi$ from $AB$ and $CD$ is equal to $k$.

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

(ROM) Given a triangle $OAB$ such that $\angle AOB=\alpha\lt 90^\circ$, let $M$ be an arbitrary point of the triangle different from $O$. Denote by $P$ and $Q$ the feet of the perpendiculars from $M$ to $OA$ and $OB$, respectively. Let $H$ be the orthocenter of the triangle $OPQ$. Find the locus of points $H$ when:
(a) $M$ belongs to the segment $AB$;
(b) $M$ belongs to the interior of $\triangle OAB$.

(POL) We are given $n\geq3$ points in the plane. Let $d$ be the maximal distance between two of the given points. Prove that the number of pairs of points whose distance is equal to $d$ is less than or equal to $n$.