MathDB

1

Part of 2020 JBMO TST of France

Problems(2)

Combinatorics game

Source: First JBMO TST of France 2020, Problem 1

3/4/2020
Players A and B play a game. They are given a box with n=>1n=>1 candies. A starts first. On a move, if in the box there are kk candies, the player chooses positive integer ll so that l<=kl<=k and (l,k)=1(l, k) =1, and eats ll candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of nn.
combinatorics
Geometry

Source: France JBMO TST 2020 Test 2 P1

3/11/2020
Given are four distinct points A,B,E,PA, B, E, P so that PP is the middle of AEAE and BB is on the segment APAP. Let k1k_1 and k2k_2 be two circles passing through AA and BB. Let t1t_1 and t2t_2 be the tangents of k1k_1 and k2k_2, respectively, to AA.Let CC be the intersection point of t2t_2 and k1k_1 and QQ be the intersection point of t2t_2 and the circumscribed circle of the triangle ECBECB. Let DD be the intersection posit of t1t_1 and k2k_2 and RR be the intersection point of t1t_1 and the circumscribed circle of triangle BDEBDE. Prove that P,Q,RP, Q, R are collinear.
geometry