2020 AIME II Problems/Problem 12
Contents
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 can vary from to . However, . Checking that value and the values around it, we can eliminate . Counting odd integers, we get .
Add all of our cases to get
-Solution by thanosaops
Solution 2 (Official MAA)
Because square is in the bottom row, it follows that . Moreover, because square is in the top row, and square is not in the top row, . In particular, because the number of rows in the rectangle must be odd, must be one of or
For each possible choice of and , let denote the line through the centers of squares and Note that for odd values of , the line passes through the center of square Thus intersects the interior of cell exactly when its slope is strictly between and . The line is vertical whenever square is the th square in the bottom row of the rectangle. This would happen for when , respectively. When is 1 greater than or 1 less than these numbers, the slope of is or , respectively. In all other cases the slope is strictly between and The admissible values for for each possible value of are given in the following table. This accounts for rectangles.
Video Solution 1
https://www.youtube.com/watch?v=MrtKoO16XLQ ~ MathEx
Video Solution 2
Video Solution 3
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
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