Really Irrational Equations

Algebra Level 5

There exist a real number \(p\) such that the following system of equations

\[\dfrac { 1 }{ \sqrt { 1+2{ x }^{ 2 } } } +\dfrac { 1 }{ \sqrt { 1+2{ y }^{ 2 } } } =\dfrac { 2 }{ \sqrt { 1+2xy } } \]

\[x\sqrt { x(1-2x) } +\sqrt { y(1-2y) } =p\]

has only one real value \(x\) as a root, which can be expressed as

\(x=\dfrac { a+\sqrt { b } }{ c } \)

where \(a, b, c\) are all non-zero integers and \(b\) is square-free. Find \(a+b+c\)

Details and Assumptions:

  • There's no typo in this question.

See Victor Loh's Irrational Equations

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