## គណិតវិទ្យា​អូឡាំពិច​អន្តរជាតិ ១៩៦៥

ប៊ែរឡាំង, អាល្លឺម៉ង់ខាង​កើត, ៣-១៣ កក្កដា ១៩៦៥

ថ្ងៃ​ទី​មួយ

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

2. (POL) Consider the system of equations
$\left \{ \begin{array}{c c} a_{11}x_1 + a_{12}x_2 + a_{13}x_{3} &= 0,\\ a_{21}x_1 + a_{22}x_2 + a_{23}x_{3} &= 0,\\ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 &= 0,\end{array} \right.$

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.

3. (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 $k$.

ថ្ងៃ​ទី​ពីរ

4. (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 $2$.

5. (ROM) Given a triangle $OAB$ such that $\angle AOB = a < 90^{\circ}$, 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 $OAB$.

6. (POL) We are given $n = 3$ 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 $n$.