DE-BUNDESWETTBEWERB 2022

Five squirrels together have a supply of 2022 nuts. On the first day 2 nuts are added, on the second day 4 nuts, on the third day 6 nuts and so on, i.e. on each further day 2 nuts more are added than on the day before. At the end of any day the squirrels divide the stock among themselves. Is it possible that they all receive the same number of nuts and that no nut is left over?

Eva draws an equilateral triangle and its altitudes. In a first step she draws the center triangle of the equilateral triangle, in a second step the center triangle of this center triangle and so on. After each step Eva counts all triangles whose sides lie completely on drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?

A circle $k$ touches a larger circle $K$ from inside in a point $P$ . Let $Q$ be point on $k$ different from $P$ . The line tangent to $k$ at $Q$ intersects $K$ in $A$ and $B$ . Show that the line $PQ$ bisects $\angle APB$ .