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Difference between revisions of "2003 AIME I Problems/Problem 2"

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== Problem ==
 
== Problem ==
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One hundred [[concentric]] [[circle]]s with [[radius | radii]] <math> 1, 2, 3, \dots, 100 </math> are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The [[ratio]] of the total [[area]] of the green regions to the area of the circle of radius 100 can be expressed as <math> m/n, </math> where <math> m </math> and <math> n </math> are [[relatively prime]] [[positive integer]]s. Find <math> m + n. </math>
+
One hundred [[concentric]] [[circle]]s with [[radius | radii]] <math> 1, 2, 3, \dots, 100 </math> are drawn in a plane. The interior of the circle of radius <math>1</math> is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The [[ratio]] of the total area of the green regions to the area of the circle of radius <math>100</math> can be expressed as <math> m/n, </math> where <math> m </math> and <math> n </math> are [[relatively prime]] [[positive integer]]s. Find <math> m + n. </math>
  
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== Solution ==
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== Solution 1 ==
−
To get the green area, we can color all the circles of radius 100 or below green, then color all those with radius 99 or below red, then color all those with radius 98 or below green, etc.  This amounts to adding the area of the circle of radius 100, but subtracting the circle of radius 99, then adding the circle of radius 98, etc.
+
To get the green area, we can color all the circles of radius <math>100</math> or below green, then color all those with radius <math>99</math> or below red, then color all those with radius <math>98</math> or below green, and so forth.  This amounts to adding the area of the circle of radius <math>100</math>, but subtracting the circle of radius <math>99</math>, then adding the circle of radius <math>98</math>, and so forth.
  
−
The total green area is thus given by <math>\pi 100^{2} - \pi 99^{2} + \pi 98^{2} - \ldots - \pi 1^{2}</math>, while the total area is given by <math>\pi 100^{2}</math>, so the ratio is
+
The total green area is thus given by <math> 100^{2} \pi - 99^{2} \pi + 98^{2} \pi - \ldots - 1^{2} \pi</math>, while the total area is given by <math>100^{2} \pi</math>, so the ratio is
−
 
+
<cmath>\frac{100^{2}\pi - 99^{2}\pi + 98^{2}\pi - \ldots - 1^{2}\pi}{100^{2}\pi}</cmath>
−
<math>\frac{\pi 100^{2} - \pi 99^{2} + \pi 98^{2} - \ldots - \pi 1^{2}}{\pi 100^{2}}</math>
 
  
 
For any <math>a</math>, <math>a^{2}-(a-1)^{2}=a^{2}-(a^{2}-2a+1)=2a-1</math>.  We can cancel the [[divisor | factor]] of [[pi]] from the [[numerator]] and [[denominator]] and simplify the ratio to  
 
For any <math>a</math>, <math>a^{2}-(a-1)^{2}=a^{2}-(a^{2}-2a+1)=2a-1</math>.  We can cancel the [[divisor | factor]] of [[pi]] from the [[numerator]] and [[denominator]] and simplify the ratio to  
  
−
<math>\frac{(2\cdot100 - 1)+(2\cdot98 - 1) + \ldots + (2\cdot 2 - 1)}{100^{2}} = \frac{2\cdot(100 + 98 + \ldots + 2) - 50}{100^2}</math>.  
+
<cmath>\frac{(2\cdot100 - 1)+(2\cdot98 - 1) + \ldots + (2\cdot 2 - 1)}{100^{2}} = \frac{2\cdot(100 + 98 + \ldots + 2) - 50}{100^2}.</cmath>   
  
 
Using the formula for the sum of an [[arithmetic series]], we see that this is equal to  
 
Using the formula for the sum of an [[arithmetic series]], we see that this is equal to  
  
−
<math>\frac{2(50)(51)-50}{100^{2}}=\frac{50(101)}{100^{2}}=\frac{101}{200}</math>,
+
<cmath>\frac{2(50)(51)-50}{100^{2}}=\frac{50(101)}{100^{2}}=\frac{101}{200},</cmath>
  
 
so the answer is <math>101 + 200 =\boxed{301}</math>.
 
so the answer is <math>101 + 200 =\boxed{301}</math>.
  
−
Alternatively, consider that in problems such as these that have obvious patterns (in this case, the radius of each circle is <math>1</math> bigger than the previous circle), the solutions usually also follow obvious patterns.  Pretend that the problem is the same except that it only involves <math>2</math> circles, then <math>4</math> circles, then on and on until a pattern emerges.
+
----
  
−
For <math>2</math> circles, the ratio is <math>3/4</math>.
+
Alternatively, we can determine a pattern through trial-and-error using smaller numbers.
−
For <math>4</math> circles, the ratio is <math>5/8</math>.
+
 
−
For <math>6</math> circles, the ratio is <math>7/12</math>.
+
*For <math>2</math> circles, the ratio is <math>3/4</math>.
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For <math>8</math> circles, the ratio is <math>9/16</math>.
+
*For <math>4</math> circles, the ratio is <math>5/8</math>.
 +
*For <math>6</math> circles, the ratio is <math>7/12</math>.
 +
*For <math>8</math> circles, the ratio is <math>9/16</math>.
  
 
Now the pattern for each ratio is clear.  Given <math>x</math> circles, the ratio is <math>\frac{x+1}{2x}</math>.
 
Now the pattern for each ratio is clear.  Given <math>x</math> circles, the ratio is <math>\frac{x+1}{2x}</math>.
 
For the <math>100</math> circle case (which is what this problem is), <math>x=100</math>, and the ratio is <math>\frac{101}{200}</math>.
 
For the <math>100</math> circle case (which is what this problem is), <math>x=100</math>, and the ratio is <math>\frac{101}{200}</math>.
  
−
So, the answer is <math>\boxed{301}</math>.
+
 
 +
----
 +
 
 +
Also, using the difference of squares, the expression simplifies to <math>\frac{100 + 99 + 98 + 97 + ... + 1}{100^2}</math>. We can easily determine the sum with <math>\frac{100(101)}{2} = 5050</math>. Simplifying gives us <math>\frac{5050}{100^2} = \frac{101}{200}</math> and the answer is <math>101 + 200 =\boxed{301}</math>.
 +
 
 +
----
 +
 
 +
== Solution 2 (synergy) ==
 +
 
 +
We want to find <math>\frac{\sum\limits_{n=1}^{50} (4n-1)\pi}{10000\pi}=\frac{\sum\limits_{n=1}^{50} (4n-1)}{10000}=\frac{(\sum\limits_{n=1}^{50} (4n) )-50}{10000}=\frac{101}{200} \rightarrow 101+200=\boxed{301}</math>
 +
 
 +
== Solution 3 (Alcumus) ==
 +
 
 +
The sum of the areas of the green regions is
 +
 
 +
<cmath>\left[(2^2-1^2)+(4^2-3^2)+(6^2-5^2)+\cdots+(100^2-99^2)\right]\pi</cmath>
 +
<cmath>=\left[(2+1)+(4+3)+(6+5)+\cdots+(100+99)\right]\pi</cmath>
 +
<cmath>={1\over2}\cdot100\cdot101\pi.</cmath>
 +
 
 +
Thus the desired ratio is<cmath>{1\over2}\cdot{{100\cdot101\pi}\over{100^2\pi}}={101\over200},</cmath>and <math>m+n=\boxed{301}</math>.
  
 
== See also ==
 
== See also ==
−
* [[2003 AIME I Problems/Problem 1 | Previous problem]]
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{{AIME box|year=2003|n=I|num-b=1|num-a=3}}
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* [[2003 AIME I Problems/Problem 3 | Next problem]]
 
−
* [[2003 AIME I Problems]]
 
  
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 14:49, 8 March 2021

Problem

One hundred concentric circles with radii $1, 2, 3, \dots, 100$ are drawn in a plane. The interior of the circle of radius $1$ is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius $100$ can be expressed as $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution 1

To get the green area, we can color all the circles of radius $100$ or below green, then color all those with radius $99$ or below red, then color all those with radius $98$ or below green, and so forth. This amounts to adding the area of the circle of radius $100$, but subtracting the circle of radius $99$, then adding the circle of radius $98$, and so forth.

The total green area is thus given by $100^{2} \pi - 99^{2} \pi + 98^{2} \pi - \ldots - 1^{2} \pi$, while the total area is given by $100^{2} \pi$, so the ratio is \[\frac{100^{2}\pi - 99^{2}\pi + 98^{2}\pi - \ldots - 1^{2}\pi}{100^{2}\pi}\]

For any $a$, $a^{2}-(a-1)^{2}=a^{2}-(a^{2}-2a+1)=2a-1$. We can cancel the factor of pi from the numerator and denominator and simplify the ratio to

\[\frac{(2\cdot100 - 1)+(2\cdot98 - 1) + \ldots + (2\cdot 2 - 1)}{100^{2}} = \frac{2\cdot(100 + 98 + \ldots + 2) - 50}{100^2}.\]

Using the formula for the sum of an arithmetic series, we see that this is equal to

\[\frac{2(50)(51)-50}{100^{2}}=\frac{50(101)}{100^{2}}=\frac{101}{200},\]

so the answer is $101 + 200 =\boxed{301}$.

Alternatively, we can determine a pattern through trial-and-error using smaller numbers.

  • For $2$ circles, the ratio is $3/4$.
  • For $4$ circles, the ratio is $5/8$.
  • For $6$ circles, the ratio is $7/12$.
  • For $8$ circles, the ratio is $9/16$.

Now the pattern for each ratio is clear. Given $x$ circles, the ratio is $\frac{x+1}{2x}$. For the $100$ circle case (which is what this problem is), $x=100$, and the ratio is $\frac{101}{200}$.


Also, using the difference of squares, the expression simplifies to $\frac{100 + 99 + 98 + 97 + ... + 1}{100^2}$. We can easily determine the sum with $\frac{100(101)}{2} = 5050$. Simplifying gives us $\frac{5050}{100^2} = \frac{101}{200}$ and the answer is $101 + 200 =\boxed{301}$.

Solution 2 (synergy)

We want to find $\frac{\sum\limits_{n=1}^{50} (4n-1)\pi}{10000\pi}=\frac{\sum\limits_{n=1}^{50} (4n-1)}{10000}=\frac{(\sum\limits_{n=1}^{50} (4n) )-50}{10000}=\frac{101}{200} \rightarrow 101+200=\boxed{301}$

Solution 3 (Alcumus)

The sum of the areas of the green regions is

\[\left[(2^2-1^2)+(4^2-3^2)+(6^2-5^2)+\cdots+(100^2-99^2)\right]\pi\] \[=\left[(2+1)+(4+3)+(6+5)+\cdots+(100+99)\right]\pi\] \[={1\over2}\cdot100\cdot101\pi.\]

Thus the desired ratio is\[{1\over2}\cdot{{100\cdot101\pi}\over{100^2\pi}}={101\over200},\]and $m+n=\boxed{301}$.

See also

2003 AIME I (Problems • Answer Key • Resources)
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