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 4 of 4: Conclude

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\frac{k^2}{s^2}\ge S_{k^2-1}\ge\frac1{r^2}\Longrightarrow s\le kr

Detailed analysis

Combining the upper bound for the final sum with its monotonic lower bound gives $k^2/s^2\ge S_{k^2-1}\ge1/r^2$ . Taking positive square roots yields $s\le kr$ , so the board contains a number not exceeding $kr$ .