Difference between revisions of "2001 AIME I Problems/Problem 3"
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== Solution 2 == | == Solution 2 == | ||
− | We find that the given equation has a <math> | + | We find that the given equation has a <math>2000^{\text{th}}</math> degree polynomial. Note that there are no multiple roots. Thus, if <math>\frac{1}{2} - x</math> is a root, <math>x</math> is also a root. Thus, we pair up <math>1000</math> pairs of roots that sum to <math>\frac{1}{2}</math> to get a sum of <math>\boxed{500}</math>. |
==Solution 3== | ==Solution 3== |
Revision as of 21:28, 18 September 2018
Problem
Find the sum of the roots, real and non-real, of the equation , given that there are no multiple roots.
Solution 1
From Vieta's formulas, in a polynomial of the form , then the sum of the roots is
.
From the Binomial Theorem, the first term of is
, but
, so the term with the largest degree is
. So we need the coefficient of that term, as well as the coefficient of
.
Applying Vieta's formulas, we find that the sum of the roots is .
Solution 2
We find that the given equation has a degree polynomial. Note that there are no multiple roots. Thus, if
is a root,
is also a root. Thus, we pair up
pairs of roots that sum to
to get a sum of
.
Solution 3
Note that if is a root, then
is a root and they sum up to
We make the substitution
so
Expanding gives
so by Vieta, the sum of the roots of
is 0. Since
has a degree of 2000, then
has 2000 roots so the sum of the roots is
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.