Difference between revisions of "2023 SSMO Tiebreaker Round Problems/Problem 2"
(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...") |
Rounak iitr (talk | contribs) (→Problem) |
||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
Let <math>P(x) = x^3 + 3ax^2 + 3bx + (a+b)</math> be a real polynomial with | Let <math>P(x) = x^3 + 3ax^2 + 3bx + (a+b)</math> be a real polynomial with | ||
| − | nonnegative and nonzero real roots <math> | + | nonnegative and nonzero real roots <math>\alpha, \beta, \gamma</math>. Suppose that <cmath>(\alpha + 1)^3 + (\beta + 1)^3 + (\gamma+1)^3 + 3P(-1) = 0.</cmath> If <math>P(1) = a_1+b_1\sqrt{c_1},</math> for squarefree <math>c_1,</math> find <math>a_1+b_1+c_1</math>. |
==Solution== | ==Solution== | ||
Latest revision as of 02:21, 31 December 2023
Problem
Let 
be a real polynomial with
nonnegative and nonzero real roots 
. Suppose that ![\[(\alpha + 1)^3 + (\beta + 1)^3 + (\gamma+1)^3 + 3P(-1) = 0.\]](http://latex.artofproblemsolving.com/2/9/c/29cce29cdd70b36ae859eaebfda309d7322b1624.png)
If 
for squarefree 
find 
.
