ប៊ែរឡាំង, អាល្លឺម៉ង់ខាងកើត, ៣-១៣ កក្កដា ១៩៦៥
1. (YUG) Find all real numbers such that
2. (POL) Consider the system of equations
whose coefficients satisfy the following conditions:
(a) 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 is the only solution to the system.
3. (CZS) A tetrahedron is given. The lengths of the edges and are and , respectively, the distance between the lines and is , and the angle between them is equal to . The tetrahedron is divided into two parts by the plane parallel to the lines and . Calculate the ratio of the volumes of the parts if the ratio between the distances of the plane from and is equal to .
4. (USS) Find four real numbers such that the sum of any of the numbers and the product of other three is equal to .
5. (ROM) Given a triangle such that , let be an arbitrary point of the triangle different from . Denote by and the feet of the perpendiculars from to and , respectively. Let be the orthocenter of the triangle . Find the locus of points when:
(a) belongs to the segment ;
(b) belongs to the interior of .
6. (POL) We are given points in the plane. Let be the maximal distance between two of the given points. Prove that the number of pairs of points whose distance is equal to is less than or equal to .