3

Part of 2010 AIME Problems

Problems(2)

Commutative Exponentiation

Source: AIME 2010I Problem 3

Suppose that y \equal{} \frac34x and x^y \equal{} y^x. The quantity x \plus{} y can be expressed as a rational number

\frac{r}{s}

r

s

are relatively prime positive integers. Find r \plus{} s.

number theoryrelatively prime

Factorial-Like Product

Source: AIME 2010II Problem 3

K

be the product of all factors (b\minus{}a) (not necessarily distinct) where

a

b

are integers satisfying

1\le a < b \le 20

. Find the greatest positive integer

n

2^n

K

factorialAMCnumber theory