2012 MPFG Problem 8

Problem

Suppose that $x$ , $y$ , and $z$ are real numbers such that $x + y + z = 3$ and $x^{2} + y^{2} + z^{2} = 6$ . What is the largest possible value of $z$ ? Express your answer in the form $a +\sqrt{b}$ , where $a$ and $b$ are positive integers.

Note

We can actually think of this question through its analytic geometric meaning/ As shown, the $1st$ equation creates a plane made by connecting the points $(3,0,0)$ , $(0,3,0)$ , and $(0,0,3)$ . The $2nd$ equation creates a sphere with radius $\sqrt{6}$ and a center at $(0,0,0)$ . The intersections of the $2$ equations create a circle. We want the maximum value of $z$ , which is obviously located on the "axis of symmetry" of the graph.

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2012 MPFG Problem 8

Problem

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