Difference between revisions of "2025 AIME I Problems/Problem 12"

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==Problem==
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The set of points in <math>3</math>-dimensional coordinate space that lie in the plane <math>x+y+z=75</math> whose coordinates satisfy the inequalities <cmath>x-yz<y-zx<z-xy</cmath>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 <math>a\sqrt{b},</math> where <math>a</math> and <math>b</math> are positive integers and <math>b</math> is not divisible by the square of any prime. Find <math>a+b.</math>
  
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==See also==
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{{AIME box|year=2025|num-b=11|num-a=13|n=I}}
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{{MAA Notice}}

Revision as of 20:07, 13 February 2025

Problem

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 $a+b.$

See also

2025 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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