# Find all positive real solutions of the system

Find all positive real solutions of the system of equations $$\begin{cases} x_1+x_2+...+x_{1994}=1994 \\ x_1^4+x_2^4+...+x_{1994}^4=x_1^3+x_2^3+....+x_{1994}^3 \end{cases}$$

''By Hölder, we have in general $$(x_1+x_2+\dots+x_n)(x_1^3+\dots+x_n^3) \le n \cdot (x_1^4+\dots+x_n^4)$$ with equality iff the $$x_i$$ are all equal. So in this case we must have $$x_1=\dots=x_{1994}=1$$.''

Theory a more algebraic solution? (sum, subtraction, etc ...)

Notice that $$x^4-x^3\geq x-1$$
with equality iff $$x=1$$. Let $$E:= x_1^4+x_2^4+...+x_{1994}^4-(x_1^3+x_2^3+....+x_{1994}^3)$$ So we have $$0=E\geq x_1+...+x_{1994}-1994=0$$
and thus all $$x_i=1$$.
• Nice solution, doesn’t seem to even use positivity of the $x_i$ +1 – Macavity Nov 30 '19 at 21:11