Difference between revisions of "2012 MPFG Problem 8"
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Suppose that <math>x</math>, <math>y</math>, and <math>z</math> are real numbers such that <math>x + y + z = 3</math> and <math>x^{2} + y^{2} + z^{2} = 6</math>. What is the largest possible value of <math>z</math>? Express your answer in the form <math>a +\sqrt{b}</math>, where <math>a</math> and <math>b</math> are positive integers. | Suppose that <math>x</math>, <math>y</math>, and <math>z</math> are real numbers such that <math>x + y + z = 3</math> and <math>x^{2} + y^{2} + z^{2} = 6</math>. What is the largest possible value of <math>z</math>? Express your answer in the form <math>a +\sqrt{b}</math>, where <math>a</math> and <math>b</math> are positive integers. | ||
| − | == | + | ==Notes== |
We can actually think of this question through its analytic geometric meaning/ As shown, the <math>1st</math> equation creates a plane made by connecting the points <math>(3,0,0)</math>, <math>(0,3,0)</math>, and <math>(0,0,3)</math>. The <math>2nd</math> equation creates a sphere with radius <math>\sqrt{6}</math> and a center at <math>(0,0,0)</math>. The intersections of the <math>2</math> equations create a circle. We want the maximum value of <math>z</math>, which is obviously located on the "axis of symmetry" of the graph. | We can actually think of this question through its analytic geometric meaning/ As shown, the <math>1st</math> equation creates a plane made by connecting the points <math>(3,0,0)</math>, <math>(0,3,0)</math>, and <math>(0,0,3)</math>. The <math>2nd</math> equation creates a sphere with radius <math>\sqrt{6}</math> and a center at <math>(0,0,0)</math>. The intersections of the <math>2</math> equations create a circle. We want the maximum value of <math>z</math>, which is obviously located on the "axis of symmetry" of the graph. | ||
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| + | ~cassphe | ||
Latest revision as of 10:42, 22 August 2025
Problem
Suppose that 
, 
, and 
are real numbers such that 
and 
. What is the largest possible value of ? Express your answer in the form 
, where 
and 
are positive integers.
Notes
We can actually think of this question through its analytic geometric meaning/ As shown, the 
equation creates a plane made by connecting the points 
, 
, and 
. The 
equation creates a sphere with radius 
and a center at 
. The intersections of the 
equations create a circle. We want the maximum value of , which is obviously located on the "axis of symmetry" of the graph.
~cassphe
