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 1== | |
− | + | The answer is <math>128</math>, achievable by <math>A=(10,0), B=(0,-63), C=(-54,1)</math>. We now show the bound. | |
− | + | We first do the following optimizations: | |
− | We | ||
− | + | -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 <math>2</math>. | |
− | |||
− | |||
− | |||
− | + | -if all of <math>A,B,C</math> lie on one side of the plane, for example <math>y>0</math>, we shift them all down, decreasing the number of moves by <math>3</math>, until one of the points is on <math>y=0</math> for the first time. | |
+ | |||
+ | Now we may assume that <math>A=(a,d)</math>, <math>B=(b,-e)</math>, <math>C=(-c,f)</math> where <math>a,b,c,d,e,f \geq 0</math>. Note we may still shift all <math>A,B,C</math> down by <math>1</math> if <math>d,f>0</math>, decreasing the number of moves by <math>1</math>, until one of <math>d,f</math> is on <math>y=0</math> for the first time. So we may assume one of <math>(a,b)</math> and <math>(d,f)</math> is <math>0</math>, by symmetry. In particular, by shoelace the answer to 2021 JMO Problem 4 is the minimum of the answers to the following problems: | ||
+ | |||
+ | |||
+ | Case 1 (where <math>a=d=0</math>) if <math>wx-yz=4042</math>, find the minimum possible value of <math>w+x+y+z</math>. | ||
+ | |||
+ | Case 2 (else) <math>wy+xy+xz=(w+x)(y+z)-wz=4042</math>, find the minimum possible value of <math>w+x+y+z</math>. | ||
+ | |||
+ | |||
+ | Note that <math>(m+n)^2=4mn+(m-n)^2</math> so if <math>m+n</math> is fixed then <math>mn</math> is maximized exactly when <math>|m-n|</math> is minimized. In particular, if <math>m+n \leq 127</math> then <math>mn-op \leq mn \leq 63*64 = 4032 <4042</math> as desired. | ||
+ | |||
+ | |||
+ | ~Lcz | ||
==See Also== | ==See Also== |
Latest revision as of 13:44, 21 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.)
Solution 1
The answer is , achievable by
. 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 .
-if all of lie on one side of the plane, for example
, we shift them all down, decreasing the number of moves by
, until one of the points is on
for the first time.
Now we may assume that ,
,
where
. Note we may still shift all
down by
if
, decreasing the number of moves by
, until one of
is on
for the first time. So we may assume one of
and
is
, by symmetry. In particular, by shoelace the answer to 2021 JMO Problem 4 is the minimum of the answers to the following problems:
Case 1 (where ) if
, find the minimum possible value of
.
Case 2 (else) , find the minimum possible value of
.
Note that so if
is fixed then
is maximized exactly when
is minimized. In particular, if
then
as desired.
~Lcz
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.