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 1 of 4: Estimate one replacement

\frac1{a^2}+\frac1{b^2}=\frac{a^2+b^2}{a^2b^2}\ge\frac{2ab}{a^2b^2}=\frac1{r^2}

Detailed analysis

For a replacement r by a,b with ab=2r^2, AM–GM gives 1/a^2+1/b^2=(a^2+b^2)/(a^2b^2)≥2ab/(a^2b^2)=1/r^2.