2021 USAJMO Problems/Problem 4

Problem

Carina has three pins, labeled $A, B$ , and $C$ , 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 $x$ and $y$ are both integers, not necessarily positive.)

Carina has three pins, labeled $A, B$ , and $C$ , 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 $x$ and $y$ are both integers, not necessarily positive.)

Solution 1 (Lcz's Solution)

We get that the answer is $128$ .

We want to make an optimization to get down to so we do WLOG, $A=(a,d)$ , $B=(b,-e)$ , $C=(-c,f)$ , where one of $a,b$ is $0$ and one of $(d,f)$ is $0$ , and $a,b,c,d,e,f \geq 0$ ,and then we do casework shoelace, which there's two cases. Case 1: where $a=d=0$ , $wx-yz=4042$ , find the minimum possible value of $w+x+y+z$ . Case 2 else or $(w+x)(y+z)-wz=4042$ , find the minimum possible value of $w+x+y+z$ . We can see that it's clear $63*64=4032<4042$ so the sum is $127$ or $(a+d)(b+c)\leq 4042$ so if the sum's less than $128$ it is impossible to get an area of a triangle greater than $2016$ . Hence the answer must be at least $128$ .

We now show that $128$ is achievable. Indeed, taking $A=(5, 1), B=(-52, 0), C=(0,-70)$ is a valid solution, so we are done.

2021 USAJMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
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2021 USAJMO Problems/Problem 4

Problem

Solution 1 (Lcz's Solution)

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