PUMaC 2025 · 团队赛 · 第 13 题

PUMaC 2025 — Team Round — Problem 13

Consider the equations x + y + z = A, 2 2 2 x + y + z = B, 3 3 3 x + y + z = C. 3 An ordered triple ( A, B, C ) ∈ R is called “nice” if the corresponding system above has exactly 2 three real solutions for x . For fixed ( A, B ) ∈ R , the set of real C such that the system is “nice” is a finite union of finite intervals; denote the sum of their lengths as ℓ ( A, B ). Find P P 5 5 ℓ ( A, B ). A =3 B =3

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