AoPS Academy
be a positive integer. Two players 
and 
play a game in which they take turns choosing positive integers 
. The rules of the game are: takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
+1 w
and 
, of equal radius intersect at different points 
and 
. Consider a circle 
externally tangent to at 
and internally tangent to at point 
. Prove that lines 
and 
intersect at a point lying on .
and be positive integers. In an 
grid, two cells are considered neighboring if they share a common edge. Kritesh performs the following actions:
in any cell of the grid.
. and .
+2 w
be non-negative real numbers such that 
Prove that
and 
of the circle with the center 
intersect at a point 
. Let 
, 
, 
, 
be tangents to this circle. Prove that the lines 
and 
are perpendicular.
is a polar of 
wrt circle and 
