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Difference between revisions of "1979 IMO Problems/Problem 5"

(Created page with "==Problem== Determine all real numbers a for which there exists positive reals <math>x_{1}, \ldots, x_{5}</math> which satisfy the relations <math> \sum_{k=1}^{5} kx_{k}=a,</m...")
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Revision as of 23:03, 29 January 2021

Problem

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $\sum_{k=1}^{5} kx_{k}=a,$ $\sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $\sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

Solution

Let $A$ be the first sum, $B$ the second and $C$ the third.

From Cauchy, we have $AC \ge B^2$. but, $AC = B^2$.

The equality of C-S occurs if the terms are proportional.

so, $jx_j/j^5x_j = ix_i/i^5x_i$ for all $i$,$j$ $\Longrightarrow i = j$, thus no such $a$ exists.

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