Really Irrational Equations

Algebra Level 5

There exist a real number pp

such that the following system of equations

11+2x2+11+2y2=21+2xy\dfrac { 1 }{ \sqrt { 1+2{ x }^{ 2 } } } +\dfrac { 1 }{ \sqrt { 1+2{ y }^{ 2 } } } =\dfrac { 2 }{ \sqrt { 1+2xy } }

xx(12x)+y(12y)=px\sqrt { x(1-2x) } +\sqrt { y(1-2y) } =p

has only one real value xx as a root, which can be expressed as

x=a+bcx=\dfrac { a+\sqrt { b } }{ c }

where a,b,ca, b, c are all non-zero integers and bb is square-free. Find a+b+ca+b+c

Details and Assumptions:

  • There's no typo in this question.

See Victor Loh's Irrational Equations

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