1997 AIME Problems/Problem 2
To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the checkerboard, or . Similarily, there are ways to pick the vertical sides, giving us rectangles.
For , there are unit squares, of the squares, and so on until of the squares. Using the sum of squares formula, that gives us .
Thus , and .
First, to find the number of squares, we can look case by case by the side length of the possible squares on the checkerboard. We see that there are ways to place a x square and for a x square. This pattern can be easily generalized and we see that the number of squares is just . This can be simplified by using the well-known formula for the sum of consecutive squares to get .
Then, to find the number of rectangles, first note that a square falls under the definition of a rectangle. We can break up the rectangles into cases for the length x width. As we note down the cases for xxxx we see they are respectively xxxx. We can quickly generalize this pattern to basically just . This gets us which is just
Now, to calculate the ratio of we divide by to get a simplified fraction of
Thus, our answer is just ~MathWhiz35
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