MathDB

15

Part of 2003 AIME Problems

Problems(2)

A difficult geometry problem

Source:

5/12/2006
In △ABC\triangle ABC, AB=360AB = 360, BC=507BC = 507, and CA=780CA = 780. Let MM be the midpoint of CA‾\overline{CA}, and let DD be the point on CA‾\overline{CA} such that BD‾\overline{BD} bisects angle ABCABC. Let FF be the point on BC‾\overline{BC} such that DF‾⊥BD‾\overline{DF} \perp \overline{BD}. Suppose that DF‾\overline{DF} meets BM‾\overline{BM} at EE. The ratio DE:EFDE: EF can be written in the form m/nm/n, where mm and nn are relatively prime positive integers. Find m+nm + n.
geometryratiotrigonometryarticlesAoPSwikigeometric transformationreflection
Beastly Polynomial

Source:

12/26/2006
Let P(x)=24x24+∑j=123(24−j)(x24−j+x24+j).P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}). Let z1,z2,…,zrz_{1},z_{2},\ldots,z_{r} be the distinct zeros of P(x),P(x), and let zk2=ak+bkiz_{k}^{2}=a_{k}+b_{k}i for k=1,2,…,r,k=1,2,\ldots,r, where i=−1,i=\sqrt{-1}, and aka_{k} and bkb_{k} are real numbers. Let ∑k=1r∣bk∣=m+np,\sum_{k=1}^{r}|b_{k}|=m+n\sqrt{p}, where m,m, n,n, and pp are integers and pp is not divisible by the square of any prime. Find m+n+p.m+n+p.
algebrapolynomialfunction2003 AIME II problem 15