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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 ...)

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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 $x_i=1$.

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  • $\begingroup$ What did you use? Chebyshev's? $\endgroup$ – Lambert macuse Nov 30 '19 at 21:09
  • $\begingroup$ Nice solution, doesn’t seem to even use positivity of the $x_i$ +1 $\endgroup$ – Macavity Nov 30 '19 at 21:11
  • $\begingroup$ no, i draw a graph and a tangent at x=1 $\endgroup$ – Aqua Nov 30 '19 at 21:11
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    $\begingroup$ Thanks for the solution! $\endgroup$ – Lambert macuse Nov 30 '19 at 21:12

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