## Integer Ladders

### Problem 309

Published on Saturday, 6th November 2010, 04:00 pm; Solved by 462; Difficulty rating: 50%In the classic "Crossing Ladders" problem, we are given the lengths `x` and `y` of two ladders resting on the opposite walls of a narrow, level street. We are also given the height `h` above the street where the two ladders cross and we are asked to find the width of the street (`w`).

Here, we are only concerned with instances where all four variables are positive integers.

For example, if `x` = 70, `y` = 119 and `h` = 30, we can calculate that `w` = 56.

In fact, for integer values `x`, `y`, `h` and 0 < `x` < `y` < 200, there are only five triplets (`x`,`y`,`h`) producing integer solutions for `w`:

(70, 119, 30), (74, 182, 21), (87, 105, 35), (100, 116, 35) and (119, 175, 40).

For integer values `x`, `y`, `h` and 0 < `x` < `y` < 1 000 000, how many triplets (`x`,`y`,`h`) produce integer solutions for `w`?