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 and are odds, and is in the top row and in the bottom, has to be , , , or . Also, taking a look at the diagram, the slope of the line connecting those centers has to have an absolute value of . Therefore, .
If , can range from to . However, divides , so looking at mods, we can easily eliminate and . Now, counting these odd integers, we get .
Similarly, let . Then can range from to . However, , so one can remove and . Counting odd integers, we get .
Take . Then, can range from to . However, , so one can verify and eliminate and . Counting odd integers, we get .
Let . Then 223249225|180022514 - 1 = 13$.
Add all of our cases to get
-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 | ||
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