4
Part of 2001 Tuymaada Olympiad
Problems(3)
The unit square is divided into 10^12 squares..
Source: Tuymaada 2001, day 1, problem 4.
4/30/2007
Unit square is divided into smaller squares (not necessarily equal). Prove that the sum of perimeters of all the smaller squares having common points with diagonal does not exceed 1500.Proposed by A. Kanel-Belov
geometryperimetercombinatorics proposedcombinatorics
Coloring of the positive reals : possible or not?
Source: Tuymaada 2001, day 2, problem 4.
4/30/2007
Is it possible to colour all positive real numbers by 10 colours so that every two numbers with decimal representations differing in one place only are of different colours? (We suppose that there is no place in a decimal representations such that all digits starting from that place are 9's.)Proposed by A. Golovanov
functioncombinatorics proposedcombinatorics
1-100 contained in union of N geometric progressions, prove N >=31
Source: Tuymaada Junior 2001 p4
4/30/2019
Natural numbers are contained in the union of geometric progressions (not necessarily with integer denominations). Prove that
geometric sequencealgebracombinatorics