2016 AMC 10B Problems/Problem 14
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line , the line and the line
The region is a right triangle which contains the following lattice points:
Squares : Suppose that the top-right corner is , with . Then to include all other corners, we need . This produces squares.
Squares : Here . To include all other corners, we need . This produces squares.
Squares : Similarly, this produces squares.
No other squares will fit in the region. Therefore the answer is .
The vertical line is just to the right of , the horizontal line is just under , and the sloped line will always be above the value of . This means they will always miss being on a coordinate with integer coordinates so you just have to count the number of squares to the left, above, and under these lines. After counting the number of , , and squares and getting , , and respectively, and we end up with .
Solution by Wwang
The endpoint lattice points are Now we split this problem into cases.
Case 1: Square has length .
The coordinates must be or and so on to The idea is that you start at and add at the endpoint, namely The number ends up being squares for this case.
Case 2: Square has length .
The coordinates must be or or and so now it starts at It ends up being
Case 3: Square has length .
The coordinates must be or so there is squares for this case.
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