341no such reals. but is this a challenge or you in search of a solution?
Find all real a for which there exist non-negative reals x_i for 1≤i≤5 satisfying the system....
\sum_{i=1}^{5}ix_i=a
\sum_{i=1}^{5}i^3x_i=a^2
\sum_{i=1}^{5}i^5x_i=a^3
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5 Answers
341
341From Cauchy-Schwarz, we get
\sum ix_i \sum i^5 x_i \ge \left(\sum i^3x_i \right)^2 \Rightarrow a^2 \ge \sum i^3x_i \right
Equality occurs when the corresponding pairs are proportional. But that is impossible in the given case and hence equality never occurs.