Problem No. 46

Algebra Level 3

If x = 2013 − y z , y = 2014 − z x x=\sqrt{2013-yz}, y=\sqrt{2014-zx} and z = 2015 − x y , z=\sqrt{2015-xy}, find ( x + y ) 2 + ( y + z ) 2 + ( z + x ) 2 (x+y)^{2}+(y+z)^{2}+(z+x)^{2}


The answer is 12084.

Benedict Dimacutac by Benedict Dimacutac , 21, Philippines

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1 solution

From the hypothesis, we get x 2 + y z = 2013 ; y 2 + x z = 2014 ; z 2 + x y = 2015 x^2+yz=2013; y^2+xz=2014; z^2+xy=2015 .

Thus,

( x + y ) 2 + ( y + z ) 2 + ( z + x ) 2 \quad(x+y)^2+(y+z)^2+(z+x)^2

= 2 ( x 2 + y 2 + z 2 + x y + y z + z x ) =2(x^2+y^2+z^2+xy+yz+zx)

= 2 ( 2013 + 2014 + 2015 ) = 12084 =2(2013+2014+2015)=\boxed{12084} .

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