[/quote]
,
and 
.![$\sqrt[3]{ab-1}+\sqrt[3]{bc-1}+\sqrt[3]{ca-1} \le \sqrt[3]{(a+b+c)^2}$](http://latex.artofproblemsolving.com/9/6/3/9633b5849794285e44519c745c1f616eceb77711.png)
is a perfect square number.
consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
.
Let 
.
be a function that satisfies the following equation for all real numbers 
:![\[
f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17.
\]](http://latex.artofproblemsolving.com/f/6/2/f625b922656061221157fdb5a3b958cfa7d57108.png)
Find the value of 
.
prime and 
a positive integer. Determine all pairs 
satisfying the equation: 
be a positive integer and let 
and 
be sets of integers satisfying the following conditions:
,
and 
,
iff 
.
and 
.
be a given integer. The Mint issues coins of 
different values 
,
is a positive integer (the number of coins of each value is unlimited). A set of values 
is called lucky, if the sum 
can be collected in a unique way (namely, by taking one coin of each value).
with 
(b) Prove that every lucky set of values 
.
non-empty subsets 
where 
are non negative integers, such that
for every 
.
Y by cubres
2008 white stones and 1 black stone are in a row.
A move consists of selecting one black stone and change the color of its neighboring stone(s).The goal is to make all stones black after a finite number of moves. Find all possible initial positions of the black stone for which this is possible.
A move consists of selecting one black stone and change the color of its neighboring stone(s).The goal is to make all stones black after a finite number of moves. Find all possible initial positions of the black stone for which this is possible.
or 
.
where 
is the number of black stones to its left. Now if we sum those values at the end we must get 
,
