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a^2 + b + c = p, a + b^2 + c = q, a + b + c^2 = r , max no of solutions

Source: Canada Repêchage 2024/6 CMOQR

March 25, 2024
algebrasystem of equations

Problem Statement

For certain real constants p,q,r p, q, r, we are given a system of equations {a2+b+c=pa+b2+c=qa+b+c2=r\begin{cases} a^2 + b + c = p \\ a + b^2 + c = q \\ a + b + c^2 = r \end{cases} What is the maximum number of solutions of real triplets (a,b,c)(a, b, c) across all possible p,q,rp, q, r? Give an example of the pp, qq, rr that achieves this maximum.
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