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1. Color of the last ball ●●●○○ 50 ptsMilan has a bag of 2020 red balls and 2021 green balls. He repeatedly draws 2 balls out of the bag uniformly at random. If they are the same color, he changes them both to the opposite color and returns them to the bag. If they are different colors, he discards them. Eventually the bag has 1 ball left. Let $p$ be the probability that it is green. Compute $\lfloor 2021 p \rfloor$.
2. Minimum density of a special set ●●●●● 200 ptsLet $S$ be a set of positive integers satisfying the following two conditions: (i) for each positive integer $n$, at least one of $n, 2n, \ldots, 100n$ is in $S$; (ii) if $a_1, a_2, b_1, b_2$ are positive integers such that $\gcd(a_1a_2, b_1b_2)=1$ and $a_1b_1, a_2b_2 \in S$, then $a_2b_1, a_1b_2 \in S$. Suppose that $S$ has natural density $r$, meaning $\lim_{n\to\infty}\frac{1}{n}|S\cap\{1,\ldots,n\}| = r$. Compute the minimum possible value of $\lfloor 10^5 r\rfloor$.