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

The answer is $128$ , achievable by $A=(10,0)$ , B=(0,-63), C=(-54,1)$. We now show the bound.

We first do the following optimizations:

-if you have a point goes both left and right, we may obviously delete both of these moves and decrease the number of moves by$ (Error compiling LaTeX. Unknown error_msg)2$.

-if all of$ (Error compiling LaTeX. Unknown error_msg)A,B,C $lie on one side of the plane, for example$ y>0 $, we shift them all down, decreasing the number of moves by$ 3 $, until one of the points is on$ y=0$for the first time.

Now we may assume that$ (Error compiling LaTeX. Unknown error_msg)A=(a,d) $,$ B=(b,-e) $,$ C=(-c,f) $where$ a,b,c,d,e,f \geq 0 $. Note we may still shift all$ A,B,C $down by$ 1 $if$ d,f>0 $, decreasing the number of moves by$ 1 $, until one of$ d,f $is on$ y=0 $for the first time. So we may assume one of$ (a,b) $and$ (d,f) $is$ 0$. In particular, by shoelace the answer to 2021 JMO Problem 4 is the minimum of the answers to the following problems:

Case 1 (where$ (Error compiling LaTeX. Unknown error_msg)a=d=0 $) if$ wx-yz=4042 $, find the minimum possible value of$ w+x+y+z $. Case 2 (else)$ wy+xy+xz=(w+x)(y+z)-wz=4042 $, find the minimum possible value of$ w+x+y+z$.

Note that$ (Error compiling LaTeX. Unknown error_msg)(m+n)^2=4mn+(m-n)^2 $so if$ m+n $is fixed then$ mn $is maximized exactly when$ |m-n| $is minimized. In particular, if$ m+n \leq 127 $then$ mn-op \leq mn \leq 63*64 = 4032 <4042$ as desired.

2021 USAJMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6
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2021 USAJMO Problems/Problem 4

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Solution 1

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