Difference between revisions of "Mock AIME 1 Pre 2005 Problems/Problem 10"
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== See also == | == See also == | ||
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[[Category:Intermediate Complex Number Problems]] | [[Category:Intermediate Complex Number Problems]] |
Revision as of 21:44, 21 February 2010
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 hexagon 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 |