សូភីយ៉ា, ប៊ុលការី ៣-១៣ កក្កដា ១៩៦៦
1. (USS) Three problems , and were given on a mathematics olympiad. All students solved at least one of these problems. The number of students who solved and not is twice the number of students who solved and not . The number of students who solved only is greater by than the number of students who along with solved at least one other problem. Among the students who solved only one problem, half solved . How many students solved only ?
2. (HUN) If , and are the sides and and the respective angles of the triangle for which , prove that the triangle is isosceles.
3. (BUL) Prove that the sum of distances from the center of the circumsphere of the regular tetrahedron to its four vertices is less than the sum of distances from any other point to the four vertices.
4. (YUG) Prove the following equality:
where and for every .
5. (CZS) Solve the following system of equations:
where a1, a2, a3, and a4 are mutually distinct real numbers.
6. (POL) Let , and be points on , and , respectively.
Prove that the area of at least one of the three triangles , and is less than or equal to one-fourth the area of