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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?

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Here's a rule of thumb: if the exponents in a problem are roughly equal to the current calendar year, then the answer to the problem is probably deceptively simple. –  David H Aug 18 at 9:04
+1 Cool problem! –  Darth Geek Aug 18 at 9:46
@DavidH: As a corollary to your rule of thumb, any exponents or other constants matching the current year also strongly suggest that the problem may be from a recent or ongoing math contest or a take-home exam. –  Ilmari Karonen Aug 18 at 23:36

2 Answers 2

up vote 26 down vote accepted

Let $e_1$, $e_2$ and $e_3$ be LHS of the the first, second and third equations respectivelly, then:

$$e_3 -2e_2 + e_1 = \\ x^{2012}(x^2 - 2x + 1) + y^{2012}(y^2-2y+1) + z^{2012}(z^2-2z + 1) = 0 \Longrightarrow \\\Longrightarrow x^{2012}(x-1)^2 + y^{2012}(y-1)^2 + z^{2012}(z-1)^2 = 0 $$

Since all the sumands are product of even powers, they cannot be negative, so they all are $0$. That means that each unknown is either $0$ or $1$. But only if they all are $1$ the original equations hold.

You can generalize this to $n$ equations of the same kind and $n$ unknowns:

$$\left\{\begin{eqnarray} &x_1^a& + &x_2^a& + &x_3^a&+&\ldots& + &x_n^a& = &n\\ &x_1^{a+1}& +& x_2^{a+1} &+&x_3^{a+1}&+& \ldots& + &x_n^{a+1}& = & n\\ &x_1^{a+2}& +& x_2^{a+2} &+&x_3^{a+2}&+& \ldots& + &x_n^{a+2}& = & n \\ & \vdots & & \vdots & & \vdots & & \ddots & & \vdots & & \vdots \\ &x_1^{a+n-1}& +& x_2^{a+n-1} &+&x_3^{a+n-1}&+& \ldots& + &x_n^{a+n-1}& = & n\end{eqnarray}\right.$$

With $a$ even and $n$ odd.

Then we wan define $e_j$ as the $LHS$ of the $j$-th equation. Then:

$$\sum_{k=1}^n \binom{n}{k}(-1)^{k+1}e_k = \sum_{k=1}^{n}x_k^a(x_k-1)^{n-1} = 0$$

So same as before, each $x_k$ is either $1$ or $0$, but only the $n$-tuple $(1,1,\cdots,1)$ satisfies the equation.

If all the RHS are $m < n$ with $m$ a positive integer, then the solutions are the $n$-tuples with $m$ ones and $n-m$ zeros permutated.

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That's clever, +1. (Comment: you should probably edit to describe each $e_j$ as the LHS if the $j^{\textrm{th}}$ equation) –  MPW Aug 18 at 9:02
@MPW Thanks. I'll do that! –  Darth Geek Aug 18 at 9:13
what a clever approach! –  hypergeometric Aug 18 at 9:43

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.

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We are lucky you did not resist ! Thank you :-) –  Claude Leibovici Aug 18 at 11:03

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