MathDB

7

Part of 2011 Saint Petersburg Mathematical Olympiad

Problems(3)

Secret Society

Source: St Petersburg Olympiad 2011, Grade 11, P7

9/15/2017
There is secret society with 20112011 members. Every member has bank account with integer balance ( can be negative). Sometimes some member give one dollar to every his friend. It is known, that after some such moves members can redistribute their money arbitrarily. Prove, that there are exactly 20102010 pairs of friends.
combinatorics
Convex quadrilateral

Source: St Petersburg Olympiad 2011, Grade 10, P7

9/15/2017
ABCDABCD - convex quadrilateral. PP is such point on ACAC and inside △ABD\triangle ABD, that ∠ACD+∠BDP=∠ACB+∠DBP=90−∠BAD\angle ACD+\angle BDP = \angle ACB+ \angle DBP = 90-\angle BAD. Prove that ∠BAD+∠BCD=90\angle BAD+ \angle BCD =90 or ∠BDA+∠CAB=90\angle BDA + \angle CAB = 90
geometry
Plays with polygon

Source: St Petersburg Olympiad 2011, Grade 9, P7

9/15/2017
Sasha and Serg plays next game with 100100-angled regular polygon . In the beggining Sasha set natural numbers in every angle. Then they make turn by turn, first turn is made by Serg. Serg turn is to take two opposite angles and add 11 to its numbers. Sasha turn is to take two neigbour angles and add 11 to its numbers. Serg want to maximize amount of odd numbers. What maximal number of odd numbers can he get no matter how Sasha plays?
combinatoricsnumber theory