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Difference between revisions of "2020 AIME II Problems/Problem 12"

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Take <math>m=7</math>. Then, <math>n</math> can range from <math>287</math> to <math>333</math>. However, <math>300|1800</math>, so one can verify and eliminate <math>299</math> and <math>301</math>. Counting odd integers, we get <math>24 - 2 = 22</math>.
 
Take <math>m=7</math>. Then, <math>n</math> can range from <math>287</math> to <math>333</math>. However, <math>300|1800</math>, so one can verify and eliminate <math>299</math> and <math>301</math>. Counting odd integers, we get <math>24 - 2 = 22</math>.
  
−
Let <math>m = 9</math>. Then <math>n can vary from </math>223<math> to </math>249<math>. However, </math>225|1800<math>. Checking that value and the values around it, we can eliminate </math>225<math>. Counting odd integers, we get </math>14 - 1 = 13$.
+
Let <math>m = 9</math>. Then <math>n</math> can vary from <math>223</math> to <math>249</math>. However, <math>225|1800</math>. Checking that value and the values around it, we can eliminate <math>225</math>. Counting odd integers, we get <math>14 - 1 = 13</math>.
  
 
Add all of our cases to get <cmath> 165+48+22+13 = \boxed{248} </cmath>
 
Add all of our cases to get <cmath> 165+48+22+13 = \boxed{248} </cmath>

Revision as of 03:01, 8 June 2020

Problem

Let $m$ and $n$ be odd integers greater than $1.$ An $m\times n$ rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers $1$ through $n$, those in the second row are numbered left to right with the integers $n + 1$ through $2n$, and so on. Square $200$ is in the top row, and square $2000$ is in the bottom row. Find the number of ordered pairs $(m,n)$ of odd integers greater than $1$ with the property that, in the $m\times n$ rectangle, the line through the centers of squares $200$ and $2000$ intersects the interior of square $1099$.

Solution

Let us take some cases. Since $m$ and $n$ are odds, and $200$ is in the top row and $2000$ in the bottom, $m$ has to be $3$, $5$, $7$, or $9$. Also, taking a look at the diagram, the slope of the line connecting those centers has to have an absolute value of $< 1$. Therefore, $m < 1800 \mod n < 1800-m$.

If $m=3$, $n$ can range from $667$ to $999$. However, $900$ divides $1800$, so looking at mods, we can easily eliminate $899$ and $901$. Now, counting these odd integers, we get $167 - 2 = 165$.

Similarly, let $m=5$. Then $n$ can range from $401$ to $499$. However, $450|1800$, so one can remove $449$ and $451$. Counting odd integers, we get $50 - 2 = 48$.

Take $m=7$. Then, $n$ can range from $287$ to $333$. However, $300|1800$, so one can verify and eliminate $299$ and $301$. Counting odd integers, we get $24 - 2 = 22$.

Let $m = 9$. Then $n$ can vary from $223$ to $249$. However, $225|1800$. Checking that value and the values around it, we can eliminate $225$. Counting odd integers, we get $14 - 1 = 13$.

Add all of our cases to get \[165+48+22+13 = \boxed{248}\]

-Solution by thanosaops

See Also

2020 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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