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HMMT February

Algebra Difficulty 5.0 AIME Find the answer

Let x,yx, y, and zz be distinct real numbers that sum to 0. Find the maximum possible value of xy+yz+zxx2+y2+z2\frac{x y+y z+z x}{x^{2}+y^{2}+z^{2}}

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Note that 0=(x+y+z)2=x2+y2+z2+2xy+2yz+2zx0=(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x. Rearranging, we get that xy+yz+zx=−12(x2+y2+z2)x y+y z+z x=-\frac{1}{2}\left(x^{2}+y^{2}+z^{2}\right), so that in fact the quantity is always equal to −1/2-1 / 2.

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