MathDB

3

Part of 2015 Saint Petersburg Mathematical Olympiad

Problems(3)

Weights sets

Source: St Petersburg Olympiad 2015, Grade 11, P3

10/17/2017
There are weights with mass 1,3,5,....,2i+1,...1,3,5,....,2i+1,... Let A(n)A(n) -is number of different sets with total mass equal nn( For example A(9)=2A(9)=2, because we have two sets 9=9=1+3+59=9=1+3+5). Prove that A(n)≤A(n+1)A(n) \leq A(n+1) for n>1n>1
combinatoricsnumber theory
Inequality with bisectors

Source: St Petersburg Olympiad 2015, Grade 10, P3

10/17/2017
ABCDABCD - convex quadrilateral. Bisectors of angles AA and DD intersect in KK, Bisectors of angles BB and CC intersect in LL. Prove 2KL≥∣AB−BC+CD−DA∣2KL \geq |AB-BC+CD-DA|
geometryinequalities
Colored table

Source: St Petersburg Olympiad 2015, Grade 9, P3

10/17/2017
All cells of 2015×20152015 \times 2015 table colored in one of 44 colors. We count number of ways to place one tetris T-block in table. Prove that T-block has cell of all 44 colors in less than 51%51\% of total number of ways.
combinatorics