2015 AMC 10A Problems/Problem 24
Problem 24
For some positive integers , there is a quadrilateral with positive integer side lengths, perimeter , right angles at and , , and . How many different values of are possible?
Solution 24
Let and be positive integers. Drop a perpendicular from to to show that, using the Pythagorean Theorem, that Simplifying yields , so . Thus, is one more than a perfect square.
The perimeter must be less than 2015. Simple calculations demonstrate that is valid, but is not. On the lower side, does not work (because ), but does work. Hence, there are 31 valid (all such that for ), and so our answer is