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 $x_i=1$.