DE-BUNDESWETTBEWERB 2017

The numbers $1,2,3,\dots,2017$ are on the blackboard. Amelie and Boris take turns removing one of those until only two numbers remain on the board. Amelie starts. If the sum of the last two numbers is divisible by $8$ , then Amelie wins. Else Boris wins. Who can force a victory?

What is the maximum number of acute interior angles a non-overlapping planar $2017$ -gon can have?

Let $M$ be the incenter of the tangential quadrilateral $A_1A_2A_3A_4$ . Let line $g_1$ through $A_1$ be perpendicular to $A_1M$ ; define $g_2,g_3$ and $g_4$ similarly. The lines $g_1,g_2,g_3$ and $g_4$ define another quadrilateral $B_1B_2B_3B_4$ having $B_1$ be the intersection of $g_1$ and $g_2$ ; similarly $B_2,B_3$ and $B_4$ are intersections of $g_2$ and $g_3$ , $g_3$ and $g_4$ , resp. $g_4$ and $g_1$ . Prove that the diagonals of quadrilateral $B_1B_2B_3B_4$ intersect in point $M$ .

$M$ $A_4$ $A_1$ $A_2$ $A_3$ $g_3$ $g_2$ $g_1$ $g_4$ $B_4$ $B_1$ $B_2$ $B_3$