MathDB

A3

Part of 2011 Princeton University Math Competition

Problems(2)

2011 PUMaC Geometry A3

Source:

9/24/2019
Let PQPQ and PRPR be tangents to a circle ω\omega with diameter ABAB so that A,Q,R,BA, Q, R, B lie on ω\omega in that order. Let HH be the projection of PP onto ABAB and let ARAR and PHPH intersect at SS. If ∠QPH=30∘\angle QPH = 30^{\circ} and ∠HPR=20∘\angle HPR = 20^\circ, find ∠ASQ\angle ASQ in degrees.
geometry
2011 PUMaC Individual Finals A3

Source:

9/24/2019
Let ABCABC be an equilateral triangle having sides of length 1, and let PP be a point in the interior of ΔABC\Delta ABC such that ∠ABP=15∘\angle ABP = 15 ^\circ. Find, with proof, the minimum possible value of AP+BP+CPAP + BP + CP. (Comment: In fact this question is incorrect, unfortunately. A more reasonable problem: Prove that AP+BP+CP≥3AP + BP + CP \ge \sqrt{3}.)
geometry