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Integers in a Triangle - AMC 10A

There is an intuitive definition of perpendicularity. It does not involve angle. Instead, it involves the notion of distance. Consider a point P and a line L not passing through it. If you wish to walk from P to L, the path of shortest distance is the perpendicular drawn from P to L. 

This idea is used in the following problem, (similar to a problem from AMC - American Mathematical Contest 10 A, 2018).

Part 1

Part 2

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