MathDB

5

Part of 2014 Saint Petersburg Mathematical Olympiad

Problems(3)

Set of natural

Source: St Petersburg Olympiad 2014, Grade 11, P5

10/24/2017
MM is infinite set of natural numbers. If a,b,a≠ba,b, a\neq b are in MM, then ab+2a^b+2 or ab−2a^b-2 ( or both) are in MM. Prove that there is composite number in MM
number theory
Some geometry

Source: St Petersburg Olympiad 2014, Grade 10, P5

10/26/2017
Incircle ω\omega of ABCABC touch ACAC at B1B_1. Point E,FE,F on the ω\omega such that ∠AEB1=∠B1FC=90\angle AEB_1=\angle B_1FC=90. Tangents to ω\omega at E,FE,F intersects in DD, and BB and DD are on different sides for line ACAC. MM- midpoint of ACAC. Prove, that AE,CF,DMAE,CF,DM intersects at one point.
geometry
Napkin on the plane

Source: St Petersburg Olympiad 2014, Grade 9, P5

10/27/2017
On a cellular plane with a cell side equal to 11, arbitrarily 100×100100 \times 100 napkin is thrown. It covers some nodes (the node lying on the border of a napkin, is also considered covered). What is the smallest number of lines (going not necessarily along grid lines) you can certainly cover all these nodes?
combinatoricsgeometry