MathDB

15

Part of 2013 Stanford Mathematics Tournament

Problems(2)

2013 General Problem 15

Source:

2/4/2013
Given regular hexagon ABCDEFABCDEF, compute the probability that a randomly chosen point inside the hexagon is inside triangle PQRPQR, where PP is the midpoint of ABAB, QQ is the midpoint of CDCD, and RR is the midpoint of EFEF.
probabilitygeometry
SMT 2013 Team #15

Source:

2/4/2013
Suppose we climb a mountain that is a cone with radius 100100 and height 44. We start at the bottom of the mountain (on the perimeter of the base of the cone), and our destination is the opposite side of the mountain, halfway up (height z=2z = 2). Our climbing speed starts at v0=2v_0=2 but gets slower at a rate inversely proportional to the distance to the mountain top (so at height zz the speed vv is (h−z)v0/h(h-z)v_0/h). Find the minimum time needed to get to the destination.
geometry3D geometryperimeter