MathDB

1

Part of 2012 JBMO ShortLists

Problems(3)

Similar with Simson's line !

Source:

2/4/2015
Let ABCABC be an equilateral triangle , and PP be a point on the circumcircle of the triangle but distinct from AA ,BB and CC. The lines through PP and parallel to BCBC , CACA , ABAB intersect the lines CACA , ABAB , BCBC at MM , NN and QQ respectively .Prove that MM , NN and QQ are collinear .
geometrycircumcircle
Name tags !

Source:

2/4/2015
Along a round table are arranged 1111 cards with the names ( all distinct ) of the 1111 members of the 16th16^{th} JBMO Problem Selection Committee . The cards are arranged in a regular polygon manner . Assume that in the first meeting of the Committee none of its 1111 members sits in front of the card with his name . Is it possible to rotate the table by some angle so that at the end at least two members sit in front of the card with their names ?
combinatorics
JBMO Shortlist 2012 N1

Source:

2/4/2015
If aa , bb are integers and s=a3+b3−60ab(a+b)≥2012s=a^3+b^3-60ab(a+b)\geq 2012 , find the least possible value of ss.