<p>The set of points in <img alt="$3$" height="12" src="https://latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png" width="8" />-dimensional coordinate space that lie in the plane <img alt="$x+y+z=75$" height="17" src="https://latex.artofproblemsolving.com/1/1/3/113f9fba9243de0fc9c4b82585189d509ff18c09.png" width="118" /> whose coordinates satisfy the inequalities<img alt="\[x-yz<y-zx<z-xy\]" height="13" src="https://latex.artofproblemsolving.com/8/2/9/82998ee0bab0aee5af1bd428ead1e25750b3a2b1.png" width="202" />forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form <img alt="$a\sqrt{b},$" height="23" src="https://latex.artofproblemsolving.com/c/1/6/c16e14124a8d9bcb06399e3fddc10ca48f9841e1.png" width="39" /> where <img alt="$a$" height="10" src="https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png" width="11" /> and <img alt="$b$" height="14" src="https://latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png" width="10" /> are positive integers and <img alt="$b$" height="14" src="https://latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png" width="10" /> is not divisible by the square of any prime. Find <img alt="$a+b.$" height="14" src="https://latex.artofproblemsolving.com/f/5/8/f588b95d2616ba65d936e39966057315d60dc23f.png" width="43" /></p>

Question No: 409

The set of points in $3$ -dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities $x-yz<y-zx<z-xy$ forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b},$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b.$

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Tags: Geometry Inequalities

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