Difference between revisions of "2003 AIME II Problems/Problem 15"
(→Problem) |
(→Solution) |
||
Line 5: | Line 5: | ||
== Solution == | == Solution == | ||
− | |||
− | |||
− | |||
− | |||
This can quite obviously be factored as: | This can quite obviously be factored as: | ||
Revision as of 18:45, 31 May 2017
Contents
[hide]Problem
Let . Let
be the distinct zeros of
and let
for
where
and
are real numbers. Let

where and
are integers and
is not divisible by the square of any prime. Find
Solution
This can quite obviously be factored as:
Note that .
So the roots of
are exactly all
-th complex roots of
, except for the root
.
Let . Then the distinct zeros of
are
.
We can clearly ignore the root as it does not contribute to the value that we need to compute.
The squares of the other roots are .
Hence we need to compute the following sum:
Using basic properties of the sine function, we can simplify this to
The five-element sum is just .
We know that
,
, and
.
Hence our sum evaluates to:
Therefore the answer is .
Solution 2
Note that . Our sum can be reformed as
So
And we can proceed as above.
See also
2003 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
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.