USAMO 2021 — Days 1 and 2

Rectangles $BCC_1B_2$, $CAA_1C_2$, and $ABB_1A_2$ are erected outside an acute triangle $ABC$. Suppose

$$\angle BC_1C+\angle CA_1A+\angle AB_1B=180^\circ.$$

Prove that $B_1C_2$, $C_1A_2$, and $A_1B_2$ are concurrent.

Proposed by Ankan Bhattacharya. Evan Chen’s third-party solution notes: https://aops.com/community/p21498558; source video https://youtu.be/9WNgDETHOlI?t=890. One point is nonofficial editorial normalization; manual essay grading only.

The Planar National Park is an undirected 3-regular planar graph: every vertex has degree three. A visitor starts at a vertex along an edge, then turns left at the next vertex, right at the following vertex, and alternates left and right thereafter. She stops on returning to her starting vertex. Over all park layouts, what is the largest number of times she could have entered a vertex?

Proposed by Zoran Sunic. Evan Chen’s third-party solution notes: https://aops.com/community/p21498640; source video https://youtu.be/9WNgDETHOlI?t=1325. One point is nonofficial editorial normalization; manual essay grading only.

Let $n\ge2$ be an integer. An $n\times n$ board is empty. Each minute one may place stones in all three empty cells of a fixed-orientation L tromino (rotations not allowed), or remove all stones from a completely filled row, or from a completely filled column. The L consists of its bottom-left, bottom-right and top-left cells.



For which $n$ can a nonempty sequence of moves return the board to empty?

Proposed by Alex Zhai, Shaunak Kishore. Evan Chen’s third-party solution notes: https://aops.com/community/p21498538; source video https://youtu.be/9WNgDETHOlI?t=2325. One point is nonofficial editorial normalization; manual essay grading only.

A finite set $S$ of positive integers has the property that for every $s\in S$ and positive divisor $d$ of $s$, exactly one $t\in S$ satisfies $\gcd(s,t)=d$; $s=t$ is allowed. Find all possible cardinalities of $S$.

Proposed by Carl Schildkraut. Evan Chen’s third-party solution notes: https://aops.com/community/p21498580; source video https://youtu.be/9WNgDETHOlI?t=1008. One point is nonofficial editorial normalization; manual essay grading only.

Let $n\ge4$. Find all positive real solutions of the cyclic system of $2n$ equations below, where indices are taken modulo $2n$:

$$\begin{aligned}a_1&=\frac1{a_{2n}}+\frac1{a_2},&a_2&=a_1+a_3,\\ a_3&=\frac1{a_2}+\frac1{a_4},&a_4&=a_3+a_5,\\ a_5&=\frac1{a_4}+\frac1{a_6},&a_6&=a_5+a_7,\\ &\vdots&&\vdots\\ a_{2n-1}&=\frac1{a_{2n-2}}+\frac1{a_{2n}},&a_{2n}&=a_{2n-1}+a_1.\end{aligned}$$

Proposed by Mohsen Jamaali. Evan Chen’s third-party solution notes: https://aops.com/community/p21498967; source video https://youtu.be/9WNgDETHOlI?t=1700. One point is nonofficial editorial normalization; manual essay grading only.

Let $ABCDEF$ be a convex hexagon with $AB\parallel DE$, $BC\parallel EF$, $CD\parallel FA$, and

$$AB\cdot DE=BC\cdot EF=CD\cdot FA.$$

Let $X,Y,Z$ be the midpoints of $AD,BE,CF$. Prove that the circumcenters of $ACE$ and $BDF$ and the orthocenter of $XYZ$ are collinear. The original statement is retained unchanged in scope; see the explicit degeneracy caveat in the solutions.

Proposed by Ankan Bhattacharya. Evan Chen’s third-party solution notes: https://aops.com/community/p21498548; source video https://youtu.be/9WNgDETHOlI?t=2930. One point is nonofficial editorial normalization; manual essay grading only.