The hardest problem on the hardest test

The video discusses the Putnam math competition, highlighting its difficulty and the elegance of solutions to its hardest problems. It presents a specific problem about the probability of a sphere's center being inside a tetrahedron formed by four random points. The video guides viewers through simplifying the problem to two dimensions, finding the probability in a circle, and then extending the solution to three dimensions. It emphasizes the importance of reframing problems and using added constructs to simplify complex questions.

What is the highest possible score one can achieve in the Putnam competition?

• 120

• 100

• 150

• 60

In the problem discussed, what shape is formed by four random points on a sphere?

• Square

• Tetrahedron

• Cube

• Pyramid

When simplifying the problem to two dimensions, what shape is formed by three random points on a circle?

• Pentagon

• Hexagon

• Triangle

• Square

What is the average probability that a triangle formed by three random points on a circle contains the center?

• 0.1

• 0.5

• 0.25

• 0.75

In the three-dimensional case, how many sections does the sphere get divided into by the planes?

• 10

• 4

• 6

• 8

What method is suggested to reframe the problem involving random points on a circle?

• Using a calculator

• Using random lines and coin flips

• Using a compass and ruler

• Using a protractor

How many equally likely outcomes are there when using the coin flip method in three dimensions?

• 6

• 4

• 8

• 10