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3 problems match “divisor function”. Clear

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1. Count positive factors of 1100 ●○○○○ 10 ptsHow many distinct positive factors does $1100$ have?
2. Minimal f(2007) for a functional equation ●●●●○ 100 ptsLet $f$ be a function taking the positive integers to the positive integers, such that $f(mf(n)) = nf(m)$ for all positive integers $m$ and $n$. Find the smallest possible value of $f(2007)$.
3. 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 $\lfloor 10^5 r\rfloor$.