MathDB

3

Part of 2010 Saint Petersburg Mathematical Olympiad

Problems(3)

Cities and roads.

Source: St Petersburg Olympiad 2010, Grade 11, P3

9/14/2017
There are 20092009 cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Minisrty destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly 7575 different cities?
combinatorics
Some geometry

Source: St Petersburg Olympiad 2010, Grade 10, P3

9/14/2017
M,NM,N are midpoints of ABAB and CDCD for convex quadrilateral ABCDABCD. Points XX and YY are on AD AD and BCBC and XD=3AX,YC=3BYXD=3AX,YC=3BY. ∠MXA=∠MYB=90\angle MXA=\angle MYB = 90. Prove that ∠XMN=∠ABC\angle XMN=\angle ABC
geometry
Irrational root

Source: St Petersburg Olympiad 2010, Grade 9, P3

9/14/2017
aa is irrational , but aa and a3−6aa^3-6a are roots of square polynomial with integer coefficients.Find aa
algebrapolynomial