TITLE: System of non-linear equations.
QUESTION [12 upvotes]: I have to find all triplets $(x,y,z)$ that satisfy:
$$x^{2012} + y^{2012} + z^{2012} = 3\\x^{2013} + y^{2013} + z^{2013} = 3\\x^{2014} + y^{2014} + z^{2014} = 3$$
I've found the trivial solution $(1,1,1)$ but I don't know how to start looking for more...
Does this system have an infinite amount of solutions?

REPLY [9 votes]: While the method already posted is nice, can't resist this hint - from the first and last equation, by the inequality of power means
$$\sqrt[2014]\frac{x^{2014}+y^{2014}+z^{2014}}3 \geqslant \sqrt[2012]\frac{x^{2012}+y^{2012}+z^{2012}}3$$
with equality iff $|x|=|y|=|z|=1$.  Now using the second equation, the unique answer is obvious. 
P.S. This obviously generalises to several variables - all you need are three equations, two of which with even exponents.