An algebra problem by Julian Yu

Algebra Level 3

If x = 2013 y z , y = 2014 x z x=\sqrt { 2013-yz } ,\quad y=\sqrt { 2014-xz } and z = 2015 x y z=\sqrt { 2015-xy } , find the value of:

( x + y ) 2 + ( y + z ) 2 + ( x + z ) 2 { (x+y) }^{ 2 }+{ (y+z) }^{ 2 }+{ (x+z) }^{ 2 }


The answer is 12084.

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1 solution

Simplifying ( x + y ) 2 + ( y + z ) 2 + ( x + z ) 2 { (x+y) }^{ 2 }+{ (y+z) }^{ 2 }+{ (x+z) }^{ 2 } we get, x 2 + y 2 + 2 x y + y 2 + z 2 + 2 y z + x 2 + z 2 + 2 x z x^2+y^2+2xy+y^2+z^2+2yz+x^2+z^2+2xz
2 ( x 2 + y 2 + z 2 + x y + y z + z x ) 2(x^2+y^2+z^2+xy+yz+zx)
●Now putting x 2 = ( 2013 y z ) x^2=(2013-yz) , y 2 = ( 2014 x z ) y^2=(2014-xz) and z 2 = ( 2015 x y ) z^2=(2015-xy) .
2 ( 2013 y z + 2014 x z + 2015 x y + x y + y z + z x 2(2013-yz+2014-xz+2015-xy+xy+yz+zx
2 ( 2013 + 2014 + 2015 ) 2(2013+2014+2015)
2 ( 6042 ) 2(6042)
12084 \large\boxed{12084}


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