Math Archive (9)

$\displaystyle f\left( x \right) =x+\int _{ 0 }^{ 1 }{ \left( { x }^{ 2 }y+{ y }^{ 2 }x \right) f\left( y \right) dy }$

$\displaystyle =x+\int _{ 0 }^{ 1 }{ \left( { x }^{ 2 }y \right) f\left( y \right) dy } +\int _{ 0 }^{ 1 }{ { y }^{ 2 }xf\left( y \right) dy }$

$\displaystyle =x+A{ x }^{ 2 }+Bx$ where $\displaystyle A=\int _{ 0 }^{ 1 }{ \left( y \right) f\left( y \right) dy } , B=\int _{ 0 }^{ 1 }{ { y }^{ 2 }f\left( y \right) dy }$

$\displaystyle \int _{ 0 }^{ 1 }{ xf\left( x \right) dx } =\int _{ 0 }^{ 1 }{ x\left( x+A{ x }^{ 2 }+Bx \right) dx } =\frac { A }{ 4 } +\frac { B+1 }{ 3 } =A$

$\displaystyle \int _{ 0 }^{ 1 }{ { x }^{ 2 }f\left( x \right) dx } =\int _{ 0 }^{ 1 }{ { x }^{ 2 }\left( x+A{ x }^{ 2 }+Bx \right) dx } =\frac { A }{ 5 } +\frac { B+1 }{ 4 } =B$

Solve simultaneous equations and we get,

$\displaystyle A=\frac { 80 }{ 119 } , B=\frac { 61 }{ 119 }$

Hence,

$\displaystyle \omega=119$

$\displaystyle \omega-2=\boxed{117}$