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How many functions \(f:\mathbb{R}\mapsto\mathbb{R}\) are there such that \[ f(x^2+y^2)=f(x+y)f(x-y) ? \]

\[\large 4y^2-22y+25=0 , \\(2y-3)^3+\frac1 {(2y-3)^3} = \ ? \]

Circumcenter:

The perpendicular bisectors of a triangle are concurrent . And the point of their concurrence is known as "Circumcenter". Basically, it is represented by '\(S\)'.

The circumcenter is equidistant from ...

\[\sin \left( \theta _{1}\right) + \sin \left( \theta _{2}\right) + \sin \left( \theta _{3}\right) =\cos \left( \theta _{1}\right) + \cos \left( \theta _{2}\right) + \cos \left( \theta _{3}\right) = 0 \\ \sin ^{2}\left( \theta _{1}\right) + \sin ^{2}\left( \theta _{2}\right) + \sin ^{2}\left( \theta _{3}\right) = \ ? \]

Let \(\omega(x)=\dfrac{ax+b}{cx+d}\), where \(a,b,c,d\) are real numbers.

Given that: \(\omega(\omega(\omega(1)))=1\) and \( \omega(\omega(\omega(2)))=2015\).

Find ...

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