2021 USAJMO

Part of USAJMO

Subcontests

(6)

1

Topsy-Turvy Triangle Trouble

An equilateral triangle

\Delta

L>0

is given. Suppose that

n

equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside

\Delta

, such that each unit equilateral triangle has sides parallel to

\Delta

, but with opposite orientation. (An example with

n=2

is drawn below.) [asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy] Prove that

n \leq \frac{2}{3} L^{2}.

1

2n equations

n \geq 4

be an integer. Find all positive real solutions to the following system of

2n

equations:
\begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\ &\vdots & &\vdots \\ a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1} \end{align*}

1

An FE. Who woulda thunk it?

\mathbb{N}

denote the set of positive integers. Find all functions

f : \mathbb{N} \rightarrow \mathbb{N}

such that for positive integers

a

b,

f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.

1

GCD Set Condition

S

of positive integers has the property that, for each

s \in S,

and each positive integer divisor

d

s

, there exists a unique element

t \in S

\text{gcd}(s, t) = d

. (The elements

s

t

could be equal.)
Given this information, find all possible values for the number of elements of

S

1

Moving P(o)in(t)s

Carina has three pins, labeled

A, B

C

, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance

1

away. What is the least number of moves that Carina can make in order for triangle

ABC

to have area 2021?
(A lattice point is a point

(x, y)

in the coordinate plane where

x

y

are both integers, not necessarily positive.)

1

Erecting Rectangles

BCC_1B_2,

CAA_1C_2,

ABB_1A_2

are erected outside an acute triangle

ABC.

\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.

Prove that lines

B_1C_2,

C_1A_2,

A_1B_2

are concurrent.