4

Part of 2015 Tournament of Towns

Problems(2)

Painting segments of a polygon Blue

Source: Tournament of Towns Spring 2015 Senior A-level

N-

gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle so that it contains two new segments outside the polygon. Prove that one can paint some of these new

2N

segments in red and the rest in blue so that the sum of lengths of all the red segments would be the same as for the blue ones. (

8

combinatoricscombinatorial geometry

Midpoints of diagonals of a Cyclic Quadrilateral

Source: Tournament of Towns Fall 2015 Senior A-level

ABCD

be a cyclic quadrilateral,

K

N

be the midpoints of the diagonals and

P

Q

be points of intersection of the extensions of the opposite sides. Prove that

\angle PKQ + \angle PNQ = 180

7

geometrycyclic quadrilateral