MathDB

2

Part of 2009 Saint Petersburg Mathematical Olympiad

Problems(3)

LCM difference

Source: St Petersburg Olympiad 2009, Grade 9, P3

8/30/2017
[x,y]−[x,z]=y−z[x,y]-[x,z]=y-z and x≠y≠z≠xx \neq y \neq z \neq x Prove, that x∣y,x∣zx|y,x|z
number theoryleast common multiple
Quadrilateral

Source: St Petersburg Olympiad 2009, Grade 10, P2

8/30/2017
ABCDABCD is convex quadrilateral with AB=CDAB=CD. ACAC and BDBD intersect in OO. X,Y,Z,TX,Y,Z,T are midpoints of BC,AD,AC,BDBC,AD,AC,BD. Prove, that circumcenter of OZTOZT lies on XYXY.
geometrycircumcircle
Choice of problem

Source: St Petersburg Olympiad 2009, Grade 9, P2

8/30/2017
There are 4040 members of jury, that want to choose problem for contest. There are list with 3030 problems. They want to find such problem, that can be solved at least half members , but not all. Every member solved 2626 problems, and every two members solved different sets of problems. Prove, that they can find problem for contest.
combinatorics