Question bank USAJMO 2021

Find all functions $f:\mathbb N\to\mathbb N$ satisfying $f(a^2+b^2)=f(a)f(b)$ and $f(a^2)=f(a)^2$ for all positive integers $a,b$. Here $\mathbb N$ denotes the positive integers.

Proposed by Vincent Huang. Evan Chen’s third-party compilation; source discussion https://aops.com/community/p21498724. Source video https://youtu.be/9WNgDETHOlI?t=63.

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.

An equilateral triangle $\Delta$ has side length $L>0$. Inside it lie $n$ equilateral triangles of side length $1$ with pairwise non-overlapping interiors. Their sides are parallel to those of $\Delta$, but they have the opposite orientation. Prove $n\le\frac23L^2$.

Proposed by Alex Zhai. Evan Chen’s third-party compilation; source discussion https://aops.com/community/p21499596. Source video https://youtu.be/9WNgDETHOlI?t=655.

Carina has three pins $A,B,C$, all at the origin of the coordinate plane. One move shifts one pin to an adjacent lattice point at distance $1$. What is the least number of moves needed for triangle $ABC$ to have area $2021$?

Proposed by Brandon Wang. Evan Chen’s third-party compilation; source discussion https://aops.com/community/p21498566. Source video https://youtu.be/9WNgDETHOlI?t=317.

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 $n\ge4$ be an integer. 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.