CAUCASUS 2021

In a triangle $ABC$ let $K$ be a point on the median $BM$ such that $CK=CM$ . It appears that $\angle CBM = 2 \angle ABM$ . Prove that $BC=MK$ .

A row of 2021 balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?