MathDB

The numbers

2, 2, ..., 2

are written on a blackboard (the number

2

is repeated

n

times). One step consists of choosing two numbers from the blackboard, denoting them as

a

and

b

, and replacing them with

\sqrt{\frac{ab + 1}{2}}

(a)

x

is the number left on the blackboard after

n - 1

applications of the above operation, prove that

x \ge \sqrt{\frac{n + 3}{n}}

(b)

Prove that there are infinitely many numbers for which the equality holds and infinitely many for which the inequality is strict.

A5