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positive real numbers satisfying 
,show that:![$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$](http://latex.artofproblemsolving.com/c/b/3/cb39e7f838c99ed8a209c6844f3d2810838d1dfd.png)
.
denote the set of natural numbers, and let 
, 
, be nine distinct tuples in 
. Show that there are three distinct elements in the set 
whose product is a perfect cube.
+2 w
are written on the board.In each step,two random numbers 
and 
are chosen and deleted.Then,the number 
is written instead.What will be the number remained on the board after the last step.
in the Cartesian Plane as follows:
for some real number 
. Denote the common area of 
and 
for some be 
. Compute the algebraic form of the function for 
.
be a polynomial with integer coefficients such that 
and 
for some integer . Prove that there exists no integer such that 
.
, Where 
has a maximum value of and a minimum value of 
in the interval ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
. Confirm whether the value of 
depends on or .
, points 
, 
, and 
lie on sides 
, 
, and 
, respectively, such that
, 
, 
, 
, 
, 
.
?
) stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.
on the whiteboard. In each move, she selects two distinct numbers, and , erases them, and replaces them with 
. She repeats this process until only one number, 
, remains. What are all the possible values of ?
, and the sum of all numbers decreases by 
as well. This value is equal to 
, and it will always be equal to 
. We add to get the sum of all numbers when there is number, and the only possible value of is 
.
Prove that
Let 
Prove that
