B2

Part of 2011 Princeton University Math Competition

Problems(3)

2011 PUMaC Algebra B2

Source:

a

b

are the roots of

x^2 - 2x + 5

|a^8 + b^8|

2011 PUMaC Number Theory B2

Source:

Two robots are programmed to communicate numbers using different bases. The first robot states: "I communicate in base 10, which interestingly is a perfect square. You communicate in base 16, which is not a perfect square." The second robot states: "I find it more interesting that the sum of our bases is the factorial of an integer." The second robot is referring to the factorial of which integer?

2011 PUMaC Individual Finals B2

Source:

Prove for irrational number

\alpha

and positive integer

n

\left( \alpha + \sqrt{\alpha^2 - 1} \right)^{1/n} + \left(\alpha - \sqrt{\alpha^2 - 1} \right)^{1/n}