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Part of 2024 Canadian Mathematical Olympiad Qualification

Problems(1)

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

For certain real constants

p, q, r

, we are given a system of equations

\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)

across all possible

p, q, r

? Give an example of the

p

q

r

that achieves this maximum.

algebrasystem of equations