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Difference between revisions of "1999 AIME Problems/Problem 8"

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== Problem ==
 
== Problem ==
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Let <math>\mathcal{T}</math> be the set of ordered triples <math>(x,y,z)</math> of nonnegative [[real number]]s that lie in the [[plane]] <math>x+y+z=1.</math>  Let us say that <math>(x,y,z)</math> supports <math>(a,b,c)</math> when exactly two of the following are true: <math>x\ge a, y\ge b, z\ge c.</math>  Let <math>\mathcal{S}</math> consist of those triples in <math>\mathcal{T}</math> that support <math>\left(\frac 12,\frac 13,\frac 16\right).</math>  The area of <math>\mathcal{S}</math> divided by the area of <math>\mathcal{T}</math> is <math>m/n,</math>  where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers, find <math>m+n.</math>
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Let <math>\mathcal{T}</math> be the set of ordered triples <math>(x,y,z)</math> of nonnegative [[real number]]s that lie in the [[plane]] <math>x+y+z=1.</math>  Let us say that <math>(x,y,z)</math> supports <math>(a,b,c)</math> when exactly two of the following are true: <math>x\ge a, y\ge b, z\ge c.</math>  Let <math>\mathcal{S}</math> consist of those triples in <math>\mathcal{T}</math> that support <math>\left(\frac 12,\frac 13,\frac 16\right).</math>  The area of <math>\mathcal{S}</math> divided by the area of <math>\mathcal{T}</math> is <math>m/n,</math>  where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.</math>
  
 
== Solution ==
 
== Solution ==
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[[Image:1999_AIME-8.png]]
 
[[Image:1999_AIME-8.png]]
  
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The region in <math>(x,y,z)</math> where <math>x \ge \frac{1}{2}, y \ge \frac{1}{3}</math> is that of a little triangle on the bottom of the above diagram, of <math>y \ge {1}{3}, z \ge \frac{1}{6}</math> is the triangle at the right, and <math>x \ge \frac 12, z \ge \frac 16</math> the triangle on the left, where the triangles are coplanar with the large equilateral triangle formed by <math>x+y+z=1,\ x,y,z \ge 0</math>. We can check that each of the three regions mentioned fall under exactly two of the inequalities and not the third.  
+
The region in <math>(x,y,z)</math> where <math>x \ge \frac{1}{2}, y \ge \frac{1}{3}</math> is that of a little triangle on the bottom of the above diagram, of <math>y \ge \frac{1}{3}, z \ge \frac{1}{6}</math> is the triangle at the right, and <math>x \ge \frac 12, z \ge \frac 16</math> the triangle on the left, where the triangles are coplanar with the large equilateral triangle formed by <math>x+y+z=1,\ x,y,z \ge 0</math>. We can check that each of the three regions mentioned fall under exactly two of the inequalities and not the third.  
  
 
[[Image:1999_AIME-8a.png]]
 
[[Image:1999_AIME-8a.png]]
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The side length of the large equilateral triangle is <math>\sqrt{2}</math>, which we can find using 45-45-90 <math>\triangle</math> with the axes. Using the formula <math>A = \frac{s^2\sqrt{3}}{4}</math> for [[equilateral triangle]]s, the area of the large triangle is <math>\frac{(\sqrt{2})^2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}</math>. Since the lines of the smaller triangles are [[parallel]] to those of the large triangle, by corresponding angles we see that all of the triangles are [[similar triangles|similar]], so they are all equilateral triangles. We can solve for their side lengths easily by subtraction, and we get <math>\frac{\sqrt{2}}{6}, \frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{2}</math>. Calculating their areas, we get <math>\frac{\sqrt{3}}{8}, \frac{\sqrt{3}}{18}, \frac{\sqrt{3}}{72}</math>. The [[ratio]] <math>\frac{\mathcal{S}}{\mathcal{T}} = \frac{\frac{9\sqrt{3} + 4\sqrt{3} + \sqrt{3}}{72}}{\frac{\sqrt{3}}{2}} = \frac{14}{36} = \frac{7}{18}</math>, and the answer is <math>m + n = \boxed{025}</math>.
 
The side length of the large equilateral triangle is <math>\sqrt{2}</math>, which we can find using 45-45-90 <math>\triangle</math> with the axes. Using the formula <math>A = \frac{s^2\sqrt{3}}{4}</math> for [[equilateral triangle]]s, the area of the large triangle is <math>\frac{(\sqrt{2})^2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}</math>. Since the lines of the smaller triangles are [[parallel]] to those of the large triangle, by corresponding angles we see that all of the triangles are [[similar triangles|similar]], so they are all equilateral triangles. We can solve for their side lengths easily by subtraction, and we get <math>\frac{\sqrt{2}}{6}, \frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{2}</math>. Calculating their areas, we get <math>\frac{\sqrt{3}}{8}, \frac{\sqrt{3}}{18}, \frac{\sqrt{3}}{72}</math>. The [[ratio]] <math>\frac{\mathcal{S}}{\mathcal{T}} = \frac{\frac{9\sqrt{3} + 4\sqrt{3} + \sqrt{3}}{72}}{\frac{\sqrt{3}}{2}} = \frac{14}{36} = \frac{7}{18}</math>, and the answer is <math>m + n = \boxed{025}</math>.
  
−
To simplify the problem, we could used the fact that the area ratios are equal to the side ratios squared, and we get <math>\left(\frac{1}{2}\right)^2 + \left(\frac{1}{3}\right^2 + \left(\frac{1}{6}\right)^2 = \frac{14}{36} = \frac{7}{18}</math>.
+
To simplify the problem, we could used the fact that the area ratios are equal to the side ratios squared, and we get <math>\left(\frac{1}{2}\right)^2 + \left(\frac{1}{3}\right)^2 + \left(\frac{1}{6}\right)^2 = \frac{14}{36} = \frac{7}{18}</math>.
  
 
== See also ==
 
== See also ==
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[[Category:Intermediate Geometry Problems]]
 
[[Category:Intermediate Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 18:40, 4 July 2013

Problem

Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$

Solution

This problem just requires a good diagram and strong 3D visualization.

1999 AIME-8.png

The region in $(x,y,z)$ where $x \ge \frac{1}{2}, y \ge \frac{1}{3}$ is that of a little triangle on the bottom of the above diagram, of $y \ge \frac{1}{3}, z \ge \frac{1}{6}$ is the triangle at the right, and $x \ge \frac 12, z \ge \frac 16$ the triangle on the left, where the triangles are coplanar with the large equilateral triangle formed by $x+y+z=1,\ x,y,z \ge 0$. We can check that each of the three regions mentioned fall under exactly two of the inequalities and not the third.

1999 AIME-8a.png

The side length of the large equilateral triangle is $\sqrt{2}$, which we can find using 45-45-90 $\triangle$ with the axes. Using the formula $A = \frac{s^2\sqrt{3}}{4}$ for equilateral triangles, the area of the large triangle is $\frac{(\sqrt{2})^2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}$. Since the lines of the smaller triangles are parallel to those of the large triangle, by corresponding angles we see that all of the triangles are similar, so they are all equilateral triangles. We can solve for their side lengths easily by subtraction, and we get $\frac{\sqrt{2}}{6}, \frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{2}$. Calculating their areas, we get $\frac{\sqrt{3}}{8}, \frac{\sqrt{3}}{18}, \frac{\sqrt{3}}{72}$. The ratio $\frac{\mathcal{S}}{\mathcal{T}} = \frac{\frac{9\sqrt{3} + 4\sqrt{3} + \sqrt{3}}{72}}{\frac{\sqrt{3}}{2}} = \frac{14}{36} = \frac{7}{18}$, and the answer is $m + n = \boxed{025}$.

To simplify the problem, we could used the fact that the area ratios are equal to the side ratios squared, and we get $\left(\frac{1}{2}\right)^2 + \left(\frac{1}{3}\right)^2 + \left(\frac{1}{6}\right)^2 = \frac{14}{36} = \frac{7}{18}$.

See also

1999 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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