1979 IMO Problems/Problem 5

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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1979 IMO Problems/Problem 5

Problem

Solution