AoPS Academy
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
+1 w
of positive integers, with 
prime, such that 
is a perfect square.
consisting of a prime number and two positive integers 
and 
such that 
and 
are both powers of .
be positive integers such that 
and 
. Let 
be the set of values attained by 
as 
runs through the positive integers. Show that is the set of all positive divisors of some positive integer.
lies inside the triangle 
and satisfies the condition 
, the point 
is such that 
is a parallelogram. Prove that 
