2005 AIME I Problems/Problem 8
Revision as of 19:50, 2 January 2022 by Fuzimiao2013 (talk | contribs) (Changed frac mn to m/n since apparently that was how it was written on the actual test)
Problem
The equation has three real roots. Given that their sum is
where
and
are relatively prime positive integers, find
Solution
Let . Then our equation reads
or
. Thus, if this equation has roots
and
, by Vieta's formulas we have
. Let the corresponding values of
be
and
. Then the previous statement says that
so taking a logarithm of that gives
and
. Thus the answer is
.
See also
2005 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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.