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Cardano Triplets

Problem 251 Published on Saturday, 20th June 2009, 09:00 am; Solved by 1039;
Difficulty rating: 70%

A triplet of positive integers (a,b,c) is called a Cardano Triplet if it satisfies the condition:

For example, (2,1,5) is a Cardano Triplet.

There exist 149 Cardano Triplets for which a+b+c ≤ 1000.

Find how many Cardano Triplets exist such that a+b+c ≤ 110,000,000.