RMO Delhi 2016
A 7-tuple \( (a_1,a_2,a_3,a_4,b_1,b_2,b_3)\) of pairwise distinct positive integers with no common factors is called a shy tuple if
- \(a_1 ^2 + a_2 ^2 + a_3^2 + a_4^2 = b_1^2 + b_2^2 + b_3^2\)
- \( a_i ^ 2 + a_j^2 \neq b_k^2 \) for all \( i, j, k \).
Find the number of Shy Tuples. If there is only tuple possible then enter your answer as the sum of all numbers in the tuple. If there are infinitely many tuples possible then enter your answer as 6666.