A2

Part of 2011 Princeton University Math Competition

Problems(2)

2011 PUMaC Algebra A2

Source:

S(m, n)

satisfies the initial conditions

S(1, n) = n

S(m, 1) = 1

, and the recurrence

S(m, n) = S(m - 1, n)S(m, n - 1)

m\geq 2, n\geq 2

. Find the largest integer

k

2^k

S(7, 7)

2011 PUMaC Individual Finals A2

Source:

Define the sequence of real numbers

\{x_n\}_{n \geq 1}

x_1

is any real number and

x_n = 1 - x_1x_2\ldots x_{n-1} \text{ for all } n > 1.

x_{2011} > \frac{2011}{2012}