Problem 1

Consider the following operation on positive real numbers written on a blackboard: choose a number

r

written on the blackboard, erase that number, and then write a pair of positive real numbers

a

and

b

satisfying

2r^2=ab

on the board. Assume that you start out with just one positive real number

r

on the blackboard, and apply this operation

k^2-1

times to end up with

k^2

positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed

kr

Step 3 of 4: Use the smallest final number

Previous step Next step

s=\min\{x_1,\ldots,x_{k^2}\},\quad \frac1{s^2}\le\frac1{x_i^2}

Detailed analysis

After the final operation, call the $k^2$ numbers $x_1,\ldots,x_{k^2}$ and let $s$ be their smallest value. Then every reciprocal square is at most $1/s^2$ , so the sum is at most $k^2/s^2$ .