MathDB

2

Part of 2015 Saint Petersburg Mathematical Olympiad

Problems(3)

Interesting primes

Source: St Petersburg Olympiad 2015, Grade 11, P2

10/17/2017
a,b>1a,b>1 - are naturals, and a2+b,a+b2a^2+b,a+b^2 are primes. Prove (ab+1,a+b)=1(ab+1,a+b)=1
number theory
Chessboard coloring

Source: St Petersburg Olympiad 2015, Grade 10, P2

10/17/2017
The beaver is chess piece that move to 22 cells by horizontal or vertical. Every cell of 100×100100 \times 100 chessboard colored in some color,such that we can not get from one cell to another with same color with one move of beaver or knight. What minimal color do we need?
combinatorics
Trapezoid geometry

Source: St Petersburg Olympiad 2015, Grade 9, P2

10/17/2017
AB=CD,AD∥BCAB=CD,AD \parallel BC and AD>BCAD>BC. Ω\Omega is circumcircle of ABCDABCD. Point EE is on Ω\Omega such that BE⊥ADBE \perp AD. Prove that AE+BC>DEAE+BC>DE
geometrycircumcircletrapezoid