Other Cool Problems 9/19/11 (from tests that I don’t care to disclose at who-knows-what-time am but still should credit them) How many perfect numbers between 1 and 100 are also divisible by 3? When the TI-34 calculator is turned upside down, some numbers still appear to be numbers. For example, 6808 appears to be 8089, but 157 doesn’t appear to be a number when turned upside down. The first number that can be read upside down as a number is 6, and the 4th such number is 60. What is the 2009th number that can be read upside down as a number? The design below is constructed by trisecting a diameter of a circle of area one and creating four semicircles as shown. Find the area of the shaded region.

Other Cool Problems 9/19/11 (Get your sleep while you still can.) Find the ratio between the area of a square inscribed in a circle and an equilateral triangle circumscribed about the same circle. In the diagram below, OR is a radius of the large circle and a diameter of the small circle. Point A lies on line OR, point P is the midpoint of segment OR, and segments AB and AC are the tangents from A to the small circle. Points B and C lie on the large circle, and segments BC and AR intersect at F. If OP = 9, what is the length of FR? B P . O F A R C