Petar's equations

Algebra Level 5

Consider the system of equations:

xa+ysin(πa2+b2)=0xb+ysin(πa2+b2)=0. \begin{aligned} xa + y\sin \left(\pi \sqrt{a^2+b^2}\right) & = 0 \\ xb + y\sin \left(\pi \sqrt{a^2+b^2}\right) &= 0 . \end{aligned}

How many different ordered pairs of integers (a,b)(a, b) subject to a<6,b<6\left| a \right| < 6, \left| b \right| < 6 are there such that the equations have a non-trivial solution (x,y)(x,y) ?

This problem is posed by Petar V.

Details and assumptions

A non-trivial solution is a pair of values (x,y)(0,0) (x,y) \neq (0,0) which satisfies both equations.


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