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Problem 5

Find all real numbers aa for which there exist non-negative real numbers x1,x2,x3,x4,x5x_1,x_2,x_3,x_4,x_5 satisfying ∑k=15kxk=a\sum_{k=1}^5kx_k=a, ∑k=15k3xk=a2\sum_{k=1}^5k^3x_k=a^2, and ∑k=15k5xk=a3\sum_{k=1}^5k^5x_k=a^3.
Step 3 of 4: Restrict a to five squares
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xk>0⟹a=k2x_k>0\Longrightarrow a=k^2
Detailed analysis

For any index with x_k positive, the square factor must vanish, so a=k^2. If all x_k are zero, the first relation gives a=0. Otherwise a is one of the five squares for k=1,2,3,4,5.