MathDB

13

Part of 2004 AIME Problems

Problems(2)

Complex roots.

Source:

11/27/2005
The polynomial P(x)=(1+x+x2+⋯+x17)2−x17P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17} has 34 complex roots of the form zk=rk[cos⁡(2πak)+isin⁡(2πak)],k=1,2,3,…,34z_k=r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34, with 0<a1≤a2≤a3≤⋯≤a34<10<a_1\le a_2\le a_3\le\cdots\le a_{34}<1 and rk>0r_k>0. Given that a1+a2+a3+a4+a5=m/na_1+a_2+a_3+a_4+a_5=m/n, where mm and nn are relatively prime positive integers, find m+nm+n.
algebrapolynomialAMCAIMEnumber theoryrelatively primegeometric series
Pentagon Problem

Source: AIME 2004 Alt. #13

11/17/2005
Let ABCDEABCDE be a convex pentagon with AB∥CEAB\parallel CE, BC∥ADBC\parallel AD, AC∥DEAC\parallel DE, ∠ABC=120∘\angle ABC=120^\circ, AB=3AB=3, BC=5BC=5, and DE=15DE=15. Given that the ratio between the area of triangle ABCABC and the area of triangle EBDEBD is m/nm/n, where mm and nn are relatively prime positive integers, find m+nm+n.
ratiogeometrynumber theoryrelatively prime