MathDB

The numbers

1

through

4^{n}

are written on a board. In each step, Pedro erases two numbers

a

and

b

from the board, and writes instead the number

\frac{ab}{\sqrt{2a^2+2b^2}}

. Pedro repeats this procedure until only one number remains. Prove that this number is less than

\frac{1}{n}

, no matter what numbers Pedro chose in each step.

2