Mock AIME 1 Pre 2005 Problems/Problem 10
Problem
is a regular heptagon inscribed in a unit circle centered at
.
is the line tangent to the circumcircle of
at
, and
is a point on
such that triangle
is isosceles. Let
denote the value of
. Determine the value of
.
Solution
Let in the complex plane ,
,
. Then the vertices of our heptagon are at points
, where
are the 7th roots of unity, ie. the complex roots of
. If
, then what we want is
. Notice that
are also the 7th roots of unity, ie. the complex roots of
. From Vieta,
is the constant term of
, or
and
is
. Thus,
.
See also
Mock AIME 1 Pre 2005 (Problems, Source) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |