Difference between revisions of "2022 AIME I Problems/Problem 15"
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==Problem== | ==Problem== | ||
| + | Let <math>x,</math> <math>y,</math> and <math>z</math> be positive real numbers satisfying the system of equations: | ||
| + | <cmath>\begin{align*} | ||
| + | \sqrt{2x-xy} + \sqrt{2y-xy} &= 1 \\ | ||
| + | \sqrt{2y-yz} + \sqrt{2z-yz} &= \sqrt2 \\ | ||
| + | \sqrt{2z-zx} + \sqrt{2x-zx} &= \sqrt3. | ||
| + | \end{align*}</cmath> | ||
| + | Then <math>\left[ (1-x)(1-y)(1-z) \right]^2</math> can be written as <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
Revision as of 19:31, 17 February 2022
Problem
Let 
and 
be positive real numbers satisfying the system of equations:
Then ![$\left[ (1-x)(1-y)(1-z) \right]^2$](http://latex.artofproblemsolving.com/e/e/1/ee1ff4ecc5f5319e77a3c7a3d7e9534a9ec343da.png)
can be written as 
where 
and 
are relatively prime positive integers. Find 
