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

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

@@ Line 1: / Line 1: @@
 == Problem ==
+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>.
+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 < 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 ==
-* [[1990 AIME Problems]]
+{{AIME box|year=1990|num-b=2|num-a=4}}
+[[Category:Intermediate Geometry Problems]]
+[[Category:Intermediate Algebra Problems]]
+{{MAA Notice}}

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