MathDB

7

Part of 2010 Saint Petersburg Mathematical Olympiad

Problems(3)

Yummy segments

Source: St Petersburg Olympiad 2010, Grade 11, P7

9/14/2017
600600 integer numbers from [1,1000][1,1000] colored in red. Natural segment [n,k][n,k] is called yummy if for every natural tt from [1,k−n][1,k-n] there are two red numbers a,ba,b from [n,k][n,k] and b−a=tb-a=t . Prove that there is yummy segment with [a,b][a,b] with b−a≥199b-a \geq 199
combinatoricsnumber theory
Coloring chess square

Source: St Petersburg Olympiad 2010, Grade 10, P7

9/14/2017
200×200200 \times 200 square is colored in chess order. In one move we can take every 2×32 \times 3 rectangle and change color of all its cells. Can we make all cells of square in same color ?
combinatoricsgeometryrectangleChessboardinvariant
Some geometry

Source: St Petersburg Olympiad 2010, Grade 9, P7

9/14/2017
Incircle of ABCABC tangent AB,AC,BCAB,AC,BC in C1,B1,A1C_1,B_1,A_1. AA1AA_1 intersect incircle in EE. NN is midpoint B1A1B_1A_1. MM is symmetric to NN relatively AA1AA_1. Prove that ∠EMC=90\angle EMC= 90
geometry