<s>[INST] Finding all the sets of three real numbers that satisfy specific equations I am having trouble finding a set of three numbers of real numbers $(x, y, z)$ satisfying $x + y + z = xy + xz + yz = 3$. I have tried factoring the equations around but I'm not having any luck and I don't think I'm going on the right path.
 [/INST] Hint:
$a+b+c=3$
$a^2+b^2+c^2=3$
EDITED
From famous Cauchy–Schwarz inequality(http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality)
You get: 
$(a^2+b^2+c^2)(c^2+a^2+b^2) \ge(ab+bc+ca)^2$
But the equality holds here,
$(a^2+b^2+c^2)^2-(ab+bc+ca)^2=0$
When we have Cauchy Schwarz inequality: 
$(a_1^2+a_2^2+..+a_n^2)(b_1^2+b_2^2+..+b_n^2) \ge (a_1b_1+a_2b_2+..a_nb_n)^2$
Equality holds iff: 
$(a_1b_2-a_2b_1)=...a_{n-1}b_{n}-a_nb_{n-1}=0$
Apply it over here you get $a^2-bc=b^2-ac=c^2-bc0 \implies a=b=c$
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