4

Part of 2011 Saint Petersburg Mathematical Olympiad

Problems(2)

Far from squares and cubes.

Source: St Petersburg Olympiad 2011, Grade 11, P4

Call integer number

x

as far from squares and cubes, if for every integer

k

|x-k^2|>10^6,|x-k^3|>10^6

. Prove, that there are infinitely many far from squares and cubes degrees of

2

Friends in city

Source: St Petersburg Olympiad 2011, Grade 10, P4

In some city there are

2000000

citizens. In every group of

2000

citizens there are

3

pairwise friends. Prove, that there are

4

pairwise friends in city.