2001 AIME I Problems/Problem 3

Revision as of 13:43, 1 December 2007 by 1=2 (talk | contribs)


Find the sum of the roots, real and non-real, of the equation $x^{2001}+\left(\frac 12-x\right)^{2001}=0$, given that there are no multiple roots.


From Vieta's formulas, we just need to find the first two terms.

From the Binomial Theorem, the first term of $left(\frac 12-x\right)^{2001}$ (Error compiling LaTeX. ! Extra \right.) is $-x^{2001}$, but $x^{2001}+-x^{2001}=0$, so the first term has $x^{2000}$ in it, not $x^{2001}$. So we find that term, and the term with $x^{1999}$.



Applying Vieta's Formulas, we get that the sum of the roots is


See also

2001 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
Invalid username
Login to AoPS