Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1990 AIME Problems/Problem 3"

(+ solution)
(→‎Solution 3)
 
(17 intermediate revisions by 10 users not shown)
Line 2: Line 2:
 
Let <math>P_1^{}</math> be a [[Regular polygon|regular]] <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that each [[interior angle]] of <math>P_1^{}</math> is <math>\frac{59}{58}</math> as large as each interior angle of <math>P_2^{}</math>. What's the largest possible value of <math>s_{}^{}</math>?
 
Let <math>P_1^{}</math> be a [[Regular polygon|regular]] <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that each [[interior angle]] of <math>P_1^{}</math> is <math>\frac{59}{58}</math> as large as each interior angle of <math>P_2^{}</math>. What's the largest possible value of <math>s_{}^{}</math>?
  
−
== Solution ==
+
== Solution 1==
 
The formula for the interior angle of a regular sided [[polygon]] is <math>\frac{(n-2)180}{n}</math>.  
 
The formula for the interior angle of a regular sided [[polygon]] is <math>\frac{(n-2)180}{n}</math>.  
  
−
Thus, <math>\frac{\frac{(r-2)180}{r}}{\frac{(s-2)180}{s}} = \frac{59}{58}</math>. Cross multiplying and simplifying, we get <math>\frac{58(r-2)}{r} = \frac{59(s-2}{s}</math>. Cross multiply and combine like terms again to yield <math>58rs - 58 \cdot 2s = 59rs - 59 \cdot 2r \Longrightarrow 118r - 116s = rs</math>. Solving for <math>r</math>, we get <math>r = \frac{116s}{118 - s}</math>.
+
Thus, <math>\frac{\frac{(r-2)180}{r}}{\frac{(s-2)180}{s}} = \frac{59}{58}</math>. Cross multiplying and simplifying, we get <math>\frac{58(r-2)}{r} = \frac{59(s-2)}{s}</math>. Cross multiply and combine like terms again to yield <math>58rs - 58 \cdot 2s = 59rs - 59 \cdot 2r \Longrightarrow 118r - 116s = rs</math>. Solving for <math>r</math>, we get <math>r = \frac{116s}{118 - s}</math>.
  
−
<math>r \ge 0</math> and <math>s \ge 0</math>, making the [[numerator]] of the [[fraction]] positive. To make the [[denominator]] [[positive]], <math>s \le 118</math>; the largest possible value of <math>s</math> is <math>117</math>.
+
<math>r \ge 0</math> and <math>s \ge 0</math>, making the [[numerator]] of the [[fraction]] positive. To make the [[denominator]] [[positive]], <math>s < 118</math>; the largest possible value of <math>s</math> is <math>117</math>.
 +
 
 +
This is achievable because the denominator is <math>1</math>, making <math>r</math> a positive number <math>116 \cdot 117</math> and <math>s = \boxed{117}</math>.
 +
 
 +
== Solution 2==
 +
Like above, use the formula for the interior angles of a regular sided [[polygon]].
 +
 
 +
 
 +
<math>\frac{(r-2)180}{r} = \frac{59}{58} * \frac{(s-2)180}{s}</math>
 +
 
 +
 
 +
<math>59 * 180 * (s-2) * r = 58 * 180 * (r-2) * s</math>
 +
 
 +
 
 +
<math>59 * (rs - 2r) = 58 * (rs - 2s)</math>
 +
 
 +
 
 +
<math>rs - 118r = -116s</math>
 +
 
 +
 
 +
<math>rs = 118r-116s</math>
 +
 
 +
 
 +
This equation tells us <math>s</math> divides <math>118r</math>. If <math>s</math> specifically divides 118 then the highest it can be is 118. However, this gives an equation with no solution. The second largest possibility in this case is <math>s=59</math>, which does give a solution: <math>s=59, r=116</math>. Although, the problem asks for <math>s</math>, not <math>r</math>. The only conceivable reasoning behind this is that <math>r</math> is greater than 1000. This prompts us to look into the second case, where <math>s</math> divides <math>r</math>. Make <math>r = s * k</math>. Rewrite the equation using this new information.
 +
 
 +
 
 +
<math>s * s * k = 118 * s * k - 116 * s</math>
 +
 
 +
 
 +
<math>s * k = 118 * k - 116</math>
 +
 
 +
 
 +
Now we now k divides 116. The larger k is, the larger s will be, so we set k to be the maximum: 116.
 +
 
 +
 
 +
<math>s * 116 = 118 * 116 - 116</math>
 +
 
 +
 
 +
<math>s = 118 - 1</math>
 +
 
 +
 
 +
<math>s = \boxed{117}</math>
 +
 
 +
 
 +
-jackshi2006
 +
 
 +
 
 +
== Solution 3 ==
 +
 
 +
As in above, we have <math>rs = 118r - 116s.</math> This means that <math>rs + 116s - 118r = 0.</math> Using SFFT we obtain <math>s(r+116) - 118(r+116) = -118 \cdot 116 \implies (s-118)(r+116) = -118 \cdot 116.</math> Since <math>r+116</math> is always positive, we know that <math>s-118</math> must be negative. Therefore the maximum value of <math>s</math> must be <math>\boxed{117}</math> which indeed yields an integral value of <math>r.</math>
 +
 
 +
== Video Solution ==
 +
 
 +
https://www.youtube.com/watch?v=9YwQlFAJqvc
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=1990|num-b=2|num-a=4}}
 
{{AIME box|year=1990|num-b=2|num-a=4}}
 +
 +
[[Category:Intermediate Geometry Problems]]
 +
[[Category:Intermediate Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 18:18, 16 April 2024

Problem

Let $P_1^{}$ be a regular $r~\mbox{gon}$ and $P_2^{}$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that each interior angle of $P_1^{}$ is $\frac{59}{58}$ as large as each interior angle of $P_2^{}$. What's the largest possible value of $s_{}^{}$?

Solution 1

The formula for the interior angle of a regular sided polygon is $\frac{(n-2)180}{n}$.

Thus, $\frac{\frac{(r-2)180}{r}}{\frac{(s-2)180}{s}} = \frac{59}{58}$. Cross multiplying and simplifying, we get $\frac{58(r-2)}{r} = \frac{59(s-2)}{s}$. Cross multiply and combine like terms again to yield $58rs - 58 \cdot 2s = 59rs - 59 \cdot 2r \Longrightarrow 118r - 116s = rs$. Solving for $r$, we get $r = \frac{116s}{118 - s}$.

$r \ge 0$ and $s \ge 0$, making the numerator of the fraction positive. To make the denominator positive, $s < 118$; the largest possible value of $s$ is $117$.

This is achievable because the denominator is $1$, making $r$ a positive number $116 \cdot 117$ and $s = \boxed{117}$.

Solution 2

Like above, use the formula for the interior angles of a regular sided polygon.


$\frac{(r-2)180}{r} = \frac{59}{58} * \frac{(s-2)180}{s}$


$59 * 180 * (s-2) * r = 58 * 180 * (r-2) * s$


$59 * (rs - 2r) = 58 * (rs - 2s)$


$rs - 118r = -116s$


$rs = 118r-116s$


This equation tells us $s$ divides $118r$. If $s$ specifically divides 118 then the highest it can be is 118. However, this gives an equation with no solution. The second largest possibility in this case is $s=59$, which does give a solution: $s=59, r=116$. Although, the problem asks for $s$, not $r$. The only conceivable reasoning behind this is that $r$ is greater than 1000. This prompts us to look into the second case, where $s$ divides $r$. Make $r = s * k$. Rewrite the equation using this new information.


$s * s * k = 118 * s * k - 116 * s$


$s * k = 118 * k - 116$


Now we now k divides 116. The larger k is, the larger s will be, so we set k to be the maximum: 116.


$s * 116 = 118 * 116 - 116$


$s = 118 - 1$


$s = \boxed{117}$


-jackshi2006


Solution 3

As in above, we have $rs = 118r - 116s.$ This means that $rs + 116s - 118r = 0.$ Using SFFT we obtain $s(r+116) - 118(r+116) = -118 \cdot 116 \implies (s-118)(r+116) = -118 \cdot 116.$ Since $r+116$ is always positive, we know that $s-118$ must be negative. Therefore the maximum value of $s$ must be $\boxed{117}$ which indeed yields an integral value of $r.$

Video Solution

https://www.youtube.com/watch?v=9YwQlFAJqvc

See also

1990 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1990_AIME_Problems/Problem_3&oldid=218261"