MathDB

14

Part of 2022 Princeton University Math Competition

Problems(1)

2022 PUMaC Team #14

Source:

9/9/2023
Let △ABC\vartriangle ABC be a triangle. Let QQ be a point in the interior of △ABC\vartriangle ABC, and let X,Y,ZX, Y,Z denote the feet of the altitudes from QQ to sides BCBC, CACA, ABAB, respectively. Suppose that BC=15BC = 15, ∠ABC=60o\angle ABC = 60^o, BZ=8BZ = 8, ZQ=6ZQ = 6, and ∠QCA=30o\angle QCA = 30^o. Let line QXQX intersect the circumcircle of △XYZ\vartriangle XY Z at the point W≠XW\ne X. If the ratio WYWZ\frac{ WY}{WZ} can be expressed as pq\frac{p}{q} for relatively prime positive integers p,qp, q, find p+qp + q.
geometryratiocircumcirclenumber theoryrelatively prime