MathDB

3

Part of 1984 National High School Mathematics League

Problems(2)

Polar Coordinates Problem

Source: 1984 National High School Mathematics League, Exam One, Problem 3

2/22/2020
For any integers 1≤n≤m≤51\leq n\leq m\leq5, how many hyperbolas does the equation ρ=11−Cmncos⁡θ\rho=\frac{1}{1-\text{C}_m^n \cos\theta} represent? Note: Cmn=m!n!(m−n)!\text{C}_m^n=\frac{m!}{n!(m-n)!}. (A)15(B)10(C)7(D)6\text{(A)}15\qquad\text{(B)}10\qquad\text{(C)}7\qquad\text{(D)}6
analytic geometryconicshyperbola
Largest Area

Source: 1984 National High School Mathematics League, Exam Two, Problem 3

2/22/2020
In △ABC\triangle ABC, PP is a point on BCBC. F∈AB,E∈AC,PF//CA,PE//BAF\in AB,E\in AC,PF//CA,PE//BA. If S△ABC=1S_{\triangle ABC}=1, prove that at least one of S△BPF,S△PCE,SPEAFS_{\triangle BPF},S_{\triangle PCE},S_{PEAF} is not less than 49\frac{4}{9}.
geometry