Math and Logic Puzzles: Redux

[geometry]

Oh yeah I just remembered I had a thing I thought of. So here's another one for anyone interested.

It turns out that if you pick three points at random (uniformly) in a unit square, the expected value of the area of the triangle with those three points as vertices is 11/144. This is a complicated result that involves mean iterated integrals to calculate. See here if you're curious.

Using this, determine the probability that two line segments with endpoints generated at random (uniformly) in a unit square will intersect.

Actually this isn't exactly true - Shoelace formula is your best friend!

[geometry]

Here's a fun one, an EGMO question slightly modified (from last year maybe, but idfk.)

Let there be finitely many lines on the plane, no three of them concurrent. Turbo the Snail (the name is crucial) begins her journey at a point on exactly one of the lines. She moves along the line in one direction until she reaches an intersection. Then she starts moving along the other line, & so on in the same fashion. At the nth intersection she encounters she turns right if n is a prime, & left otherwise. Is there a segment that Turbo the Snail passes through in both directions?

european girls math olympiad?

[geometry]

here's a geometry question (quite easy):

Consider circle W and points A,B outside circle W. Let the tangents from A to W intersect W at P,Q and let the midpoint of PQ be A'. Define B' similarly. Prove that AA'B'B is cyclic.