B3

Part of 2012 Princeton University Math Competition

Problems(4)

2012 PUMaC Geometry B3

Source:

Let A be a regular

12

-sided polygon. A new

12

-gon B is constructed by connecting the midpoints of the sides of A. The ratio of the area of B to the area of A can be written in simplest form as

(a +\sqrt{b})/c

a, b, c

are integers. Find

a + b + c

2012 PUMaC Algebra B3

Source:

\sqrt[3]{26 + 15\sqrt3} + \sqrt[3]{26 - 15\sqrt3}

2012 PUMaC Number Theory B3

Source:

How many factors of

(20^{12})^2

20^{12}

are not factors of

20^{12}

2012 PUMaC Individual Finals B3

Source:

Find, with proof, all pairs

(x, y)

of integers satisfying the equation

3x^2+ 4 = 2y^3