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Almost right-angled triangles I

Problem 223 Published on Friday, 26th December 2008, 05:00 pm; Solved by 1233;
Difficulty rating: 70%

Let us call an integer sided triangle with sides abc barely acute if the sides satisfy
a2 + b2 = c2 + 1.

How many barely acute triangles are there with perimeter ≤ 25,000,000?