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== | + | 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)== | ||
| + | We get that the answer is <math>128</math>. | ||
| + | |||
| + | 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. | ||
| + | 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>. | ||
| + | Case 2 else or <math>(w+x)(y+z)-wz=4042</math>, find the minimum possible value of <math>w+x+y+z</math>. | ||
| + | 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 16:41, 15 April 2021
Problem
Carina has three pins, labeled 
, and 
, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance 
away. What is the least number of moves that Carina can make in order for triangle 
to have area 2021?
(A lattice point is a point 
in the coordinate plane where 
and 
are both integers, not necessarily positive.)
Carina has three pins, labeled , and , respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance away. What is the least number of moves that Carina can make in order for triangle to have area 2021?
(A lattice point is a point in the coordinate plane where and are both integers, not necessarily positive.)
Solution 1 (Lcz's Solution)
We get that the answer is 
.
We want to make an optimization to get down to so we do WLOG, 
, 
, 
, where one of 
is 
and one of 
is , and 
,and then we do casework shoelace, which there's two cases.
Case 1: where 
, 
, find the minimum possible value of 
.
Case 2 else or 
, find the minimum possible value of .
We can see that it's clear 
so the sum is 
or (a+d)(b+c)/leq 
so if the sum's less than it is impossible to get an area of a triangle greater than 
. Thus done.
See Also
| 2021 USAJMO (Problems • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAJMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
