<s>[INST] Find numbers $a, b, c$ given that $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$ 
Let $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$. Find $a,b,c$

Suppose $a, b, c$ are roots of $P(x)$. 
$$P(x) = k(x - a)(x - b)(x - c)$$
But then I get $(k = 1)$
$$P(x) = x^3 - 12x^2 + x(ab + ac + bc) - abc$$
Cant go further...
 [/INST] $$a^2+b^2+c^2=(a+b+c)^2-2(ab+ac+bc)$$
So $ab+ac+bc=\frac{(a+b+c)^2-(a^2+b^2+c^2)}{2}=47$
Similarly 
$$a^3+b^3+c^3=(a+b+c)^3+3abc-3(a+b+c)(ab+ac+bc)$$
So $abc=\frac{a^3+b^3+c^3+3(a+b+c)(ab+ac+bc)-(a+b+c)^3}{3}=44$
And $a,b,c$ are solutions of the cubic
$$X^3-12X^2+47X-44=0$$
No simple solution except Cardan formulaes 
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