3

Part of 2007 Estonia National Olympiad

Problems(4)

headteacher hires new teachers, algebra problem

Source: 2007 Estonia National Olympiad Final Round grade 9 p3

The headteacher wants to hire a certain number of new teachers in addition to existing teachers. If he hired an additional

10

teachers, the number of school students would be reduced number per teacher by

5

. However, if the headmaster hired

20

new teachers, the number of students per teacher would be reduced by

8

. How many students and how many there are teachers in this school? https://cdn.artofproblemsolving.com/attachments/2/8/c0157ff43fd3d92138c87556a0fca2414e8a3f.png

Estonian Math Competitions 2006/2007

Source: Finalround Problem 4

Prove that the sum of the squares of any three pairwise different positive odd integers can be represented as the sum of the squares of six (not necessarily different) positive integers.

number theory unsolvednumber theory

Estonian Math Competitions 2006/2007

Source: Finalround Problem 9

A circle passing through the endpoints of the leg AB of an isosceles triangle ABC intersects the base BC in point P. A line tangent to the circle in point B intersects the circumcircle of ABC in point Q. Prove that P lies on line AQ if and only if AQ and BC are perpendicular.

geometrycircumcirclegeometry unsolved

Estonian Math Competitions 2006/2007

Source: Finalround Problem 14

Does there exist an equilateral triangle (a) on a plane; (b) in a 3-dimensional space; such that all its three vertices have integral coordinates?

calculusintegrationanalytic geometrygeometry unsolvedgeometry