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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 2 of 4: Use non-negativity
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∑k=15k(a−k2)2xk=0\sum_{k=1}^5 k(a-k^2)^2x_k=0
Detailed analysis

Every summand is non-negative because k is positive and x_k is non-negative. Since their sum is zero, every summand must itself be zero.