<s>[INST] System of simultaneous equations $x^n+y^n+z^n=3$ ($n$=1,2,5) Does someone know the solution of the following system of simultaneous equations ($x$, $y$, $z$ are real numbers):
$$
x+y+z=3
$$
$$
x^2+y^2+z^2=3
$$
$$
x^5+y^5+z^5=3
$$
Presented at one of math competitions, don't remember when and where.
 [/INST] Here is the algebraic solution. Consider z as parameter, so we have
$$
x+y=3-z
$$
$$
x^2+y^2=3-z^2
$$
Subtracting second equation from first squared gives:
$$
2xy=6-6z+2z^2
$$
or
$$
xy=z^2-3z+3
$$
Roots of the system
$$
x+y=3-z
$$
$$
xy=z^2-3z+3
$$
are the roots of polynomial
$$
t^2-(3-z)t+(z^2-3z+3)
$$
which has real roots if
$$
(3-z)^2-4(z^2-3z+3) \ge 0
$$
Simplifying the LHS expression gives:
$$
-3(z-1)^2 \ge 0
$$
or
$$
(z-1)^2 \le 0
$$
which may happen only if $z=1$.
Substituting $z=1$ gives $x=y=1$
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