Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2023 SSMO Tiebreaker Round Problems/Problem 2

Revision as of 21:30, 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...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_SSMO_Tiebreaker_Round_Problems/Problem_2&oldid=207689"