HMMT February

Algebra Difficulty 4.8 AIME Find the answer

If $x, y$ , and $z$ are real numbers such that $2 x^{2}+y^{2}+z^{2}=2 x-4 y+2 x z-5$ , find the maximum possible value of $x-y+z$ .

Your answer

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

Solution

The equation rearranges as $(x-1)^{2}+(y+2)^{2}+(x-z)^{2}=0$ , so we must have $x=1$ , $y=-2, z=1$ , giving us 4 .

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