User:Wsjradha/Cotangent Sum Problem
Cotangent Sum Problem
AIME-level of difficulty
Let be the twenty (complex) roots of the equation:
Calculate the value of:
For the purpose of this solution will be the sum of the roots of the 20th degree polynomial, taken at a time. For example,
Also, will be the sum of the cotangent inverses of the roots, taken at a time. The cotangent inverses will be multiplied as necessary, then added.
Also, will be the sum of the tangents of the cotangent inverses of the roots, taken at a time. Basically, this is the same as except that the tangents are taken right after the cotangent inverses. For example,
Let This equals There is a formula that states the following, where, for the purposes of this formula only, , is the sum of through , taken at a time, in the fashion described above:
When applied to this problem, it yields:
Taking the reciprocal of either side, one gets:
Multiple the numerator and the denominator of the right hand side by .
can be determined, from the original 20th degree equation using Vieta's Formulas, to be Therefore,
This simplifies to