2023 SSMO Tiebreaker Round Problems/Problem 2

Problem

Let $P(x) = x^3 + 3ax^2 + 3bx + (a+b)$ be a real polynomial with nonnegative and nonzero real roots $\alpha, \beta, \gamma$ . Suppose that $(\alpha + 1)^3 + (\beta + 1)^3 + (\gamma+1)^3 + 3P(-1) = 0.$ If $P(1) = a_1+b_1\sqrt{c_1},$ for squarefree $c_1,$ find $a_1+b_1+c_1$ .

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2023 SSMO Tiebreaker Round Problems/Problem 2

Problem

Solution