# [Solutions] United States of America Ersatz Mathematical Olympiad 2020

1. 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 $x$, $y$, $z$?
2. 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?
3. 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 $OH$.
4. A function $f$ from the set of positive real numbers to itself satisfies $$f(x + f(y) + xy) = xf(y) + f(x + y)$$for all positive real numbers $x$ and $y$. Prove that $f(x) = x$ for all positive real numbers $x$.
5. The sides of a convex $200$-gon $A_1 A_2 \dots A_{200}$ are colored red and blue in an alternating fashion. Suppose the extensions of the red sides determine a regular $100$-gon, as do the extensions of the blue sides. Prove that the $50$ diagonals $\overline{A_1A_{101}},\ \overline{A_3A_{103}}, \dots, \overline{A_{99}A_{199}}$ are concurrent.
6. Prove that for every odd integer $n > 1$, there exist integers $a, b > 0$ such that, if we let $Q(x) = (x + a)^2 + b$, then the following conditions hold
• we have $\gcd(a, n) = \gcd(b, n) = 1$;
• the number $Q(0)$ is divisible by $n$; and
• the numbers $Q(1), Q(2), Q(3), \dots$ each have a prime factor not dividing $n$.
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