Difference between revisions of "2024 AMC 8 Problems/Problem 23"

(Video Solution 2 by OmegaLearn.org)
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==Video Solution 2 by OmegaLearn.org==
 
==Video Solution 2 by OmegaLearn.org==
 
https://youtu.be/wNymnFQfN_k
 
https://youtu.be/wNymnFQfN_k
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 +
==Video Solution by SpreadTheMathLove==
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https://www.youtube.com/watch?v=x8Zo7QOB-us

Revision as of 13:05, 26 January 2024

Problem

Rodrigo is drawing lines on the coordinate plane, and counting how many unit squares they go through. He draws a line with endpoints $(2000,3000)$ and $(5000,8000).$ How many unit squares does this segment go through?

Solution 1

Let $f(x, y)$ be the number of cells the line segment from $(0, 0)$ to $(x, y)$ passes through. The problem is then equivalent to finding \[f(5000-2000, 8000-3000)=f(3000, 5000).\] Sometimes the segment passes through lattice points in between the endpoints, which happens $\text{gcd}(3000, 5000)-1=999$ times. This partitions the segment into $1000$ congruent pieces that pass through $f(3, 5)$ cells, which means the answer is \[1000f(3, 5).\] Note that a new square is entered when the lines pass through one of the lines in the coordinate grid, which for $f(3, 5)$ happens $3-1+5-1=6$ times. Because $3$ and $5$ are relatively prime, no lattice point except for the endpoints intersects the line segment from $(0, 0)$ to $(3, 5).$ This means that including the first cell closest to $(0, 0),$ The segment passes through $f(3, 5)=6+1=7$ cells. Thus, the answer is $\boxed{7000}.$ Alternatively, $f(3, 5)$ can be found by drawing an accurate diagram, leaving you with the same answer.

~BS2012

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

https://youtu.be/wNymnFQfN_k

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=x8Zo7QOB-us