2020 AIME II Problems/Problem 12

Problem

Let and be odd integers greater than An rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers through , those in the second row are numbered left to right with the integers through , and so on. Square is in the top row, and square is in the bottom row. Find the number of ordered pairs of odd integers greater than with the property that, in the rectangle, the line through the centers of squares and intersects the interior of square .

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. Therefore, .

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 .

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 .

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 .

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 .

Add all of our cases to get

-Solution by thanosaops

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.