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

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Let <math>m</math> and <math>n</math> be odd integers greater than <math>1.</math> An <math>m\times n</math> rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers <math>1</math> through <math>n</math>, those in the second row are numbered left to right with the integers <math>n + 1</math> through <math>2n</math>, and so on. Square <math>200</math> is in the top row, and square <math>2000</math> is in the bottom row. Find the number of ordered pairs <math>(m,n)</math> of odd integers greater than <math>1</math> with the property that, in the <math>m\times n</math> rectangle, the line through the centers of squares <math>200</math> and <math>2000</math> intersects the interior of square <math>1099</math>
 
Let <math>m</math> and <math>n</math> be odd integers greater than <math>1.</math> An <math>m\times n</math> rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers <math>1</math> through <math>n</math>, those in the second row are numbered left to right with the integers <math>n + 1</math> through <math>2n</math>, and so on. Square <math>200</math> is in the top row, and square <math>2000</math> is in the bottom row. Find the number of ordered pairs <math>(m,n)</math> of odd integers greater than <math>1</math> with the property that, in the <math>m\times n</math> rectangle, the line through the centers of squares <math>200</math> and <math>2000</math> intersects the interior of square <math>1099</math>
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==Solution==
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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.
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If m is 3, n can be 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.
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Similarly, let m be 5. Then n can range from 401 to 499. However, 450 divides 1800, so one can remove 449 and 451. Counting odd integers, we get 50 - 2 = 48.
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Take m as 7. Then, n can range from 287 to 333. However, 300 divides 1800, so one can verify and eliminate 299 and 301. Counting odd integers, we get 24 - 2 = 22.
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Let m be 9. Then n can vary from 223 to 249. However, 225 divides 1800. Checking that value and the values around it, we can eliminate 225. Counting odd integers, we get 14 - 1 = 13.
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Add all off our cases to get <cmath> 165+48+22+13 = \boxed{248} </cmath>

Revision as of 15:08, 7 June 2020

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 is 3, n can be 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 be 5. Then n can range from 401 to 499. However, 450 divides 1800, so one can remove 449 and 451. Counting odd integers, we get 50 - 2 = 48.

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

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

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

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