AoPS Academy
consisting of a prime number 
and two positive integers 
and 
such that 
and 
are both powers of .
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.)
,we get 
. Now just with a simple example we can see that 
goes to infinity. Take 
to get smaller than 
,but sufficiently large sum which can be higher 
.
,we can use the AM-GM theorem, to have ![$3 \geq ab+bc+ca \geq 3\sqrt[3]{a^2b^2c^2}$](http://latex.artofproblemsolving.com/9/0/b/90bb6793e43c09da96db20110614929a5d33cc6f.png)
.So, 
.With the example below,we can get that limit of goes to zero in positive numbers:
and 
and let 
be reals such that 
satisfying
Prove
Find the range of values that can be taken by :
are positive reals.
and 
Prove that
and ![$ab+bc+ca \geq 3\sqrt[3]{a^2b^2c^2}$](http://latex.artofproblemsolving.com/f/1/1/f115c481a345f3ef53d5e39d4cca4248e32aab63.png)
finishes.