MathDB

3

Part of 2014 Ukraine National Mathematical Olympiad

Problems(2)

Natural numbers satisfying an inequality

Source: Ukrainian National Math Olympiad 4th round

10/20/2014
It is known that for natural numbers a,b,c,da,b,c,d and nn the following inequalities hold: a+c<na+c<n and a/b+c/d<1a/b+c/d<1. Prove that a/b+c/d<1āˆ’1/n3a/b+c/d<1-1/n^3.
inequalitiesnumber theory unsolvednumber theory
Points and planes in space

Source: Ukrainian National Math Olympiad 4th round

10/20/2014
Is it possible to choose 2424 points in the space,such that no three of them lie on the same line and choose 20132013 planes in such a way that each plane passes through at least 33 of the chosen points and each triple of points belongs to at least one of the chosen planes?
combinatorics unsolvedcombinatorics