<s>[INST] Find $p$ for which all solutions of system/equation are real There is system of $5$ equations
$$
a+b+c+d+e = p; \\
a^2+b^2+c^2+d^2+e^2 = p; \\
a^3+b^3+c^3+d^3+e^3 = p; \\
a^4+b^4+c^4+d^4+e^4 = p; \\
a^5+b^5+c^5+d^5+e^5 = p, \\
\tag{1}
$$
where $p\in\mathbb{R}$.
One can prove (like here) that $a,b,c,d,e$ are roots of equation
$$
x^5-\binom{p}{1}x^4+\binom{p}{2}x^3-\binom{p}{3}x^2 + \binom{p}{4}x - \binom{p}{5} = 0.
\tag{2}
$$
I want to create related task for my son: to find polynomial $(2)$ for given system $(1)$. But it would be great to create task, where all $a,b,c,d,e$ are real and distinct. To have possibility to check manually all  the sums (without complex numbers).
Question: Which values of $p$ provide $5$ real (pairwise) distinct $a,b,c,d,e$? And if there exists such one at all (for $5$ variables)?
 [/INST] One can also use inequalities to give some restrictions on the solution. 
We have $$ \sum a^2(a-1)^2 = \sum a^2 - 2 \sum a^3 + \sum a^4 = p - 2p + p = 0 $$ any solution $(a,b,c,d,e)$. Since $x^2(x-1)^2 \ge 0$ for all real $x$, each of $a, b, c, d, e$ must be either $0$ or $1$. Obviously, they cannot be all distinct. 
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