MathLabs

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 4 of 4: Verify existence and include zero
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a∈{1,4,9,16,25}a\in\{1,4,9,16,25\}
Detailed analysis

For each k=1,2,3,4,5, choose x_k=k and all other variables zero; then the three relations hold with a=k^2. The all-zero choice gives a=0. Therefore the complete set is {0,1,4,9,16,25}.