G285 2021 MC-IME II

Problem 1

Let $20$ points lie in the interior of the circumcircle of regular decagon $ABCDEFGHIJ$. If each point is connected to the second farthest point away from itself by a line, find the maximum number of regions created by the intersections of lines.

Solution

Problem 2

Suppose $\phi(n)$ denote the number of integer $k$ where $1 \le k < n$ such that $\gcd(k,n)=1$. If \[\sum_{p=1}^{n} \frac{1}{2^p \phi(p)}\] can be represented as $\phi(n) \cdot l$ for some constant $l$, find $l \mod 1000$

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