AUT_ABooklet_2023

The number of subsets of $\{1,2,\dots ,2n\}$ that have $k$ as smallest element is $2^{2n-k}$ for $1\le k\le 2n$ since each element bigger than $k$ is either contained in the subset or not. The number $O$ of subsets with an odd smallest element is therefore equal to $$O=2^{2n-1}+2^{2n-3}+\cdots +2^3+2^1=2\cdot (4^{n-1}+4^{n-2}+\cdots +4^1+4^0).$$ The number $E$ of subsets with an even smallest element is equal to $$E=2^{2n-2}+2^{2n-4}+\cdots +2^2+2^0=4^{n-1}+4^{n-2}+\cdots +4^1+4^0.$$ This implies $O=2E$ and consequently the desired proportion is $2/3$. (Birgit Vera Schmidt) $\square$