Question

Who here is really good at mathematics and wants to feel like a Harvard/MIT College student?

Answer 1

Why would some of us want to go back to those days? Well, okay, there are good reasons, but they mostly don't involve doing problem sets! :-)

Answer 2

(No claim is made whatsoever about being really good at mathematics, btw)

Answer 3

Let C be the curve obtained by intersecting a cylinder of radius R and a plane. Insert two spheres of radius R into the cylinder above and below the plane, and let F1 and F2 be the points where the plane is tangent to the spheres. Let K be the vertical distance between the equators of the two spheres. Rediscover Archimedes's proof that C is an ellipse by showing that every point P on C satisfies:\[PF1+PF2=K\]Hint: If two lines through a opint P are tangent to a sphere and intersect the sphere at Q1 and Q2, then the setments PW1 and PQ2 have equal length. Use this to show that PF1 = PR1 and PF2=PR2