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2023 SSMO Tiebreaker Round Problems/Problem 2

Revision as of 22:30, 15 December 2023 by Pinkpig (talk | contribs) (Created page with "==Problem== Let <math>P(x) = x^3 + 3ax^2 + 3bx + (a+b)</math> be a real polynomial with nonnegative and nonzero real roots <math>p, q, r</math>. Suppose that <cmath>(p + 1)^3...")
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Problem

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

Solution

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