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Number of solution of a system of equations

Source: IMOC 2023 N6

September 9, 2023
algebrasystem of equations

Problem Statement

Let S(b)S(b) be the number of nonuples of positive integers (a1,a2,…,a9)(a_1, a_2, \ldots , a_9) satisfying 3b−1=a1+a2+…+a93b-1=a_1+a_2+\ldots+a_9 and b2+1=a12+…+a92b^2+1=a_1^2+\ldots+a_9^2. Prove that for all ϵ>0\epsilon>0, there exists Cϵ>0C_{\epsilon}>0 such that S(b)≤Cϵb3+ϵS(b)\leq C_{\epsilon}b^{3+\epsilon}.
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