USEMO 2020

Which positive integers can be written in the form $$\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}$$ for positive integers $x$ , $y$ , $z$ ?

Calvin and Hobbes play a game. First, Hobbes picks a family $F$ of subsets of $\{1, 2, . . . , 2020\}$ , known to both players. Then, Calvin and Hobbes take turns choosing a number from $\{1, 2, . . . , 2020\}$ which is not already chosen, with Calvin going first, until all numbers are taken (i.e., each player has $1010$ numbers). Calvin wins if he has chosen all the elements of some member of $F$ , otherwise Hobbes wins. What is the largest possible size of a family $F$ that Hobbes could pick while still having a winning strategy?

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$ . Let $\Gamma$ denote the circumcircle of triangle $ABC$ , and $N$ the midpoint of $OH$ . The tangents to $\Gamma$ at $B$ and $C$ , and the line through $H$ perpendicular to line $AN$ , determine a triangle whose circumcircle we denote by $\omega_A$ . Define $\omega_B$ and $\omega_C$ similarly. Prove that the common chords of $\omega_A$ , $\omega_B$ and $\omega_C$ are concurrent on line $OH$ .

Proposed by Anant Mudgal