Difference between revisions of "2021 USAJMO Problems/Problem 4"

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(A lattice point is a point <math>(x, y)</math> in the coordinate plane where <math>x</math> and <math>y</math> are both integers, not necessarily positive.)
 
(A lattice point is a point <math>(x, y)</math> in the coordinate plane where <math>x</math> and <math>y</math> are both integers, not necessarily positive.)
  
==Solution==
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Carina has three pins, labeled <math>A, B</math>, and <math>C</math>, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance <math>1</math> away. What is the least number of moves that Carina can make in order for triangle <math>ABC</math> to have area 2021?
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(A lattice point is a point <math>(x, y)</math> in the coordinate plane where <math>x</math> and <math>y</math> are both integers, not necessarily positive.)
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==Solution 1 (Lcz's Solution)==
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We get that the answer is <math>128</math>.
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We want to make an optimization to get down to so we do WLOG, <math>A=(a,d)</math>, <math>B=(b,-e)</math>, <math>C=(-c,f)</math>, where one of <math>a,b</math> is <math>0</math> and one of <math>(d,f)</math> is <math>0</math>, and <math>a,b,c,d,e,f \geq 0</math>,and then we do casework shoelace, which there's two cases.
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Case 1: where <math>a=d=0</math>,  <math>wx-yz=4042</math>, find the minimum possible value of <math>w+x+y+z</math>.
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Case 2 else or <math>(w+x)(y+z)-wz=4042</math>, find the minimum possible value of <math>w+x+y+z</math>.
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We can see that it's clear <math>63*64=4032<4042</math> so the sum is <math>127</math> or (a+d)(b+c)/leq <math>4042</math> so if the sum's less than <math>128</math> it is impossible to get an area of a triangle greater than <math>2016</math>. Thus done.
  
 
==See Also==
 
==See Also==

Revision as of 17:41, 15 April 2021

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 $2016$. Thus done.

See Also

2021 USAJMO (ProblemsResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6
All USAJMO Problems and Solutions

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