Difference between revisions of "2024 AMC 8 Problems/Problem 23"
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| + | Note: A general form for finding <math>f(x, y)</math> is <math>x+y-\text{gcd}(x, y).</math> We subtract <math>\text{gcd}(x, y)</math> to account for overlapping, when the line segment goes through a lattice point. | ||
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| + | ~mathkiddus | ||
==Video Solution 1 by Math-X (First fully understand the problem!!!)== | ==Video Solution 1 by Math-X (First fully understand the problem!!!)== | ||
Revision as of 18:31, 26 January 2024
Contents
Problem
Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point 
to point 
and colors the 
cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point 
to point 
. How many cells will he color this time?
![[asy] draw((-1,5)--(-1,-1),gray(.8)); draw((0,5)--(0,-1),gray(.8)); draw((1,5)--(1,-1),gray(.8)); draw((2,5)--(2,-1),gray(.8)); draw((3,5)--(3,-1),gray(.8)); draw((4,5)--(4,-1),gray(.8)); draw((5,5)--(5,-1),gray(.8)); draw((-1,5)--(5, 5),gray(.8)); draw((-1,4)--(5,4),gray(.8)); draw((-1,3)--(5,3),gray(.8)); draw((-1,2)--(5,2),gray(.8)); draw((-1,1)--(5,1),gray(.8)); draw((-1,0)--(5,0),gray(.8)); draw((-1,-1)--(5,-1),gray(.8)); dot((0,4)); label("$(0,4)$",(0,4),NW); dot((2,0)); label("$(2,0)$",(2,0),SE); draw((0,4)--(2,0)); draw((-1,0) -- (5,0), arrow=Arrow); draw((0,-1) -- (0,5), arrow=Arrow); [/asy]](http://latex.artofproblemsolving.com/4/c/d/4cd34a427b903dbb01f4e9fc1402ff0ccc15a74c.png)
Solution 1
Let 
be the number of cells the line segment from 
to 
passes through. The problem is then equivalent to finding ![\[f(5000-2000, 8000-3000)=f(3000, 5000).\]](http://latex.artofproblemsolving.com/2/3/3/233aea4231046445b3386df8788ad35801395647.png)
Sometimes the segment passes through lattice points in between the endpoints, which happens 
times. This partitions the segment into 
congruent pieces that pass through 
cells, which means the answer is ![\[1000f(3, 5).\]](http://latex.artofproblemsolving.com/4/a/1/4a1d83d64d185b9c9d6c52569e18986cc39a7b09.png)
Note that a new square is entered when the lines pass through one of the lines in the coordinate grid, which for happens 
times. Because 
and 
are relatively prime, no lattice point except for the endpoints intersects the line segment from to 
This means that including the first cell closest to 
The segment passes through 
cells. Thus, the answer is 
Alternatively, can be found by drawing an accurate diagram, leaving you with the same answer.
~BS2012
Note: A general form for finding is 
We subtract 
to account for overlapping, when the line segment goes through a lattice point.
~mathkiddus
Video Solution 1 by Math-X (First fully understand the problem!!!)
https://www.youtube.com/watch?v=dqqAk-Cd_5M
~Math-X
Video Solution 2 by OmegaLearn.org
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=x8Zo7QOB-us
