satisfying the following conditions:
divides 
,
divides 
,
.
and 
. Prove that
be positive reals such that 
.![\[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]](http://latex.artofproblemsolving.com/3/b/d/3bd44cede90261ab949beabd32f8c08411ddc051.png)
such that for all 
,![\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]](http://latex.artofproblemsolving.com/d/1/0/d10c01d7d416243614b24f04218d157e8b025ff2.png)
Y by
A chess king was placed on a square of an 
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.)
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.)
if the distance from center is the same after the move.
will result with more distance after the move call 
.
.
(divide 
to 
so the king need to cross each part and come back).
(other distance definitions are also possible).Then the distance can change -2,-1,0,1 or 2.Then for nonnegative integers 
we have 
and 
and our goal is to maximize 
.
to maximize the value.Then we have 
and 
.
we clearly have no solution and for 
and 
we can have 
and 
.
