Difference between revisions of "1999 AIME Problems/Problem 8"
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== Problem == | == Problem == | ||
| + | Let <math>\displaystyle \mathcal{T}</math> be the set of ordered triples <math>\displaystyle (x,y,z)</math> of nonnegative real numbers that lie in the plane <math>\displaystyle x+y+z=1.</math> Let us say that <math>\displaystyle (x,y,z)</math> supports <math>\displaystyle (a,b,c)</math> when exactly two of the following are true: <math>\displaystyle x\ge a, y\ge b, z\ge c.</math> Let <math>\displaystyle \mathcal{S}</math> consist of those triples in <math>\displaystyle \mathcal{T}</math> that support <math>\displaystyle \left(\frac 12,\frac 13,\frac 16\right).</math> The area of <math>\displaystyle \mathcal{S}</math> divided by the area of <math>\displaystyle \mathcal{T}</math> is <math>\displaystyle m/n,</math> where <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers, find <math>\displaystyle m+n.</math> | ||
== Solution == | == Solution == | ||
== See also == | == See also == | ||
| + | * [[1999_AIME_Problems/Problem_7|Previous Problem]] | ||
| + | * [[1999_AIME_Problems/Problem_9|Next Problem]] | ||
* [[1999 AIME Problems]] | * [[1999 AIME Problems]] | ||
Revision as of 00:53, 22 January 2007
Problem
Let 
be the set of ordered triples 
of nonnegative real numbers that lie in the plane 
Let us say that supports 
when exactly two of the following are true: 
Let 
consist of those triples in that support 
The area of divided by the area of is 
where 
and 
are relatively prime positive integers, find 
