MathDB

3

Part of 2017 Serbia National Math Olympiad

Problems(2)

Combinatorics, Serbia MO 2017

Source: Serbia National Mathematical Olympiad 2017, Day 1, Problem 3

4/1/2017
There are 2n−12n-1 lamps in a row. In the beginning only the middle one is on (the nn-th one), and the other ones are off. Allowed move is to take two non-adjacent lamps which are turned off such that all lamps between them are turned on and switch all of their states from on to off and vice versa. What is the maximal number of moves until the process terminates?
combinatorics
Tangents inducing isogonals

Source: Serbian MO 2017 6

4/2/2017
Let kk be the circumcircle of △ABC\triangle ABC and let kak_a be A-excircle .Let the two common tangents of k,kak,k_a cut BCBC in P,QP,Q.Prove that ∡PAB=∡CAQ\measuredangle PAB=\measuredangle CAQ.
geometryexcircle