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Difference between revisions of "2000 AIME II Problems/Problem 15"

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We can make an approximation by observing the following points:
 
We can make an approximation by observing the following points:
  
−
The average term is around the 60's which gives 4/3.
+
The average term is around the 60's which gives <math>\frac{4}{3}</math>.
  
−
There are 45 terms so the approximate sum is 60.
+
There are 45 terms, so the approximate sum is 60.
  
−
Therefore the entire thing equals approximately <math>\frac{1}{60}</math>.
+
Therefore, <math>\sin(n^\circ)</math> equals approximately <math>\frac{1}{60}</math>.
  
−
Recall that the approximation of sinx in radians is x if x is close to zero. In this case x is close to zero. Converting to radians we see that sin1 in degrees is about sin<math>\frac{1}{57}</math> in radians, or is about <math>\frac{1}{57}</math> because of the approximation. What we want is apparently close to that so we make the guess that n is equal to 1 degree. Basically, it boils down to the approximation of sin1=<math>\frac{1}{60}</math> in degrees, convert to radians and use the small angle approximation sinx=x.
+
Recall that the approximation of <math>\sin(x)</math> in radians is x if x is close to zero. In this case x is close to zero. Converting to radians we see that <math>\sin(1)</math> in degrees is about sin<math>\frac{1}{57}</math> in radians, or is about <math>\frac{1}{57}</math> because of the approximation. What we want is apparently close to that so we make the guess that n is equal to 1 degree. Basically, it boils down to the approximation of <math>\sin(1)=\frac{1}{60}</math> in degrees, convert to radians and use the small angle approximation <math>\sin(x)=x</math>.
 +
 
 +
~edited for clarity by fermat_sLastAMC
 +
 
 +
== Solution 3 (Alternate Finish) ==
 +
Let S be the sum of the sequence. We begin the same as in Solution 1 to get
 +
<math>S\sin(1)=\cot(45)-\cot(46)+\cot(47)-\cot(48)+...+\cot(133)-\cot(134)</math>. Observe that this "almost telescopes," if only we had some extra terms. Consider adding the sequence <math>\frac{1}{\sin(46)\sin(47)}+\frac{1}{\sin(48)\sin(49)}+...+\frac{1}{\sin(134)\sin(135)}</math>. By the identity <math>\sin(x)=\sin(180-x)</math>, this sequence is equal to the original one, simply written backwards. By the same logic as before, we may rewrite this sequence as
 +
<math>S\sin(1)=\cot(46)-\cot(47)+\cot(48)-\cot(49)+...+\cot(134)-\cot(135)</math>,
 +
and when we add the two sequences, they telescope to give <math>2S\sin(1)=\cot(45)-\cot(135)=2</math>.
 +
Hence, <math>S=\frac{1}{\sin(1^\circ)}</math>, and our angle is <math>\boxed{001}</math>.
 +
 
 +
~keeper1098
 +
 
 +
== Solution 4 ==
 +
First, multiply <math>\sin n^{\circ}</math> on both sides.
 +
<cmath>
 +
\begin{align*}
 +
    \frac{\sin n^\circ}{\sin m^\circ \sin (m+1)^\circ}
 +
    &= \frac{\sin (k+n-k)^\circ}{\sin m^\circ \sin (m+1)^\circ} \\
 +
    &= \frac{\sin (k+n)^\circ \cos k^\circ - \sin k^\circ \cos (k+n)^\circ}{\sin m^\circ \sin (m+1)^\circ}
 +
\end{align*}
 +
</cmath>
 +
Let <math>k = m</math> since <math>k</math> is could be any number.
 +
<cmath>
 +
\begin{align*}
 +
    &\quad \ \frac{\sin (k+n)^\circ \cos k^\circ}{\sin m^\circ \sin (m+1)^\circ} - \frac{\sin k^\circ \cos (k+n)^\circ}{\sin m^\circ \sin (m+1)^\circ} \\[0.5em]
 +
    &=\frac{\sin (m+n)^\circ \cos m^\circ}{\sin m^\circ \sin (m+1)^\circ} - \frac{\sin m^\circ \cos (m+n)^\circ}{\sin m^\circ \sin (m+1)^\circ}
 +
\end{align*}
 +
</cmath>
 +
 
 +
===Lemma===
 +
    <math>n</math> is equal to <math>1</math>.
 +
 
 +
===Proof===
 +
<cmath>
 +
\begin{align*}
 +
    &=\frac{\sin (m+1)^\circ \cos m^\circ}{\sin m^\circ \sin (m+1)^\circ} - \frac{\sin m^\circ \cos (m+1)^\circ}{\sin m^\circ \sin (m+1)^\circ} \\[0.5em]
 +
    &=\frac{\cos m^\circ}{\sin m^\circ} - \frac{\cos (m+1)^\circ}{\sin (m+1)^\circ} \\[0.5em]
 +
    &= \cot m^\circ - \cot (m+1)^\circ
 +
\end{align*}
 +
</cmath>
 +
The sum of all numbers could be written. Moreover, notice that <math>\cot\alpha + \cot\beta = 0</math> if <math>\alpha + \beta = 180^\circ</math>.
 +
<cmath>
 +
\begin{align*}
 +
    &\quad \ \cot 45^\circ - \cot 46^\circ + \cot 47^\circ - \cot 48^\circ + \cdots - \cot 132^\circ + \cot 133^\circ - \cot 134^\circ \\
 +
    &= (\cot 45^\circ + \cot 47^\circ + \cdots + \cot 89^\circ + \cot 91^\circ + \dots + \cot 133^\circ) \\
 +
    &\qquad\qquad\qquad\qquad\qquad\qquad - (\cot 46^\circ + \dots + \cot 88^\circ + \cot 90^\circ + \cot 92^\circ + \dots + \cot 134^\circ) \\
 +
    &= \cot 45^\circ - \cot 90^\circ \\
 +
    &= 1
 +
\end{align*}
 +
</cmath>
 +
Because <math>1 = 1</math>, the lemma is true.
 +
 
 +
Q.E.D.
 +
 
 +
<math>n</math> could be 1. Moreover, there are no smaller positive integer less than 1 to test. Thus, the least positive integer <math>n</math> that satisfies the given condition is <math>\boxed{001}</math>.
 +
 
 +
~MaPhyCom
  
 
== See also ==
 
== See also ==

Latest revision as of 00:58, 24 June 2025

Problem

Find the least positive integer $n$ such that

$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$

Solution 1

We apply the identity

\begin{align*} \frac{1}{\sin n \sin (n+1)} &= \frac{1}{\sin 1} \cdot \frac{\sin (n+1) \cos n - \sin n \cos (n+1)}{\sin n \sin (n+1)} \\ &= \frac{1}{\sin 1} \cdot \left(\frac{\cos n}{\sin n} - \frac{\cos (n+1)}{\sin (n+1)}\right) \\ &= \frac{1}{\sin 1} \cdot \left(\cot n - \cot (n+1)\right). \end{align*}

The motivation for this identity arises from the need to decompose those fractions, possibly into telescoping.

Thus our summation becomes

\[\sum_{k=23}^{67} \frac{1}{\sin (2k-1) \sin 2k} = \frac{1}{\sin 1} \left(\cot 45 - \cot 46 + \cot 47 - \cdots + \cot 133 - \cot 134 \right).\]

Since $\cot (180 - x) = - \cot x$, the summation simply reduces to $\frac{1}{\sin 1} \cdot \left( \cot 45 - \cot 90 \right) = \frac{1 - 0}{\sin 1} = \frac{1}{\sin 1^{\circ}}$. Therefore, the answer is $\boxed{001}$.

Solution 2

We can make an approximation by observing the following points:

The average term is around the 60's which gives $\frac{4}{3}$.

There are 45 terms, so the approximate sum is 60.

Therefore, $\sin(n^\circ)$ equals approximately $\frac{1}{60}$.

Recall that the approximation of $\sin(x)$ in radians is x if x is close to zero. In this case x is close to zero. Converting to radians we see that $\sin(1)$ in degrees is about sin$\frac{1}{57}$ in radians, or is about $\frac{1}{57}$ because of the approximation. What we want is apparently close to that so we make the guess that n is equal to 1 degree. Basically, it boils down to the approximation of $\sin(1)=\frac{1}{60}$ in degrees, convert to radians and use the small angle approximation $\sin(x)=x$.

~edited for clarity by fermat_sLastAMC

Solution 3 (Alternate Finish)

Let S be the sum of the sequence. We begin the same as in Solution 1 to get $S\sin(1)=\cot(45)-\cot(46)+\cot(47)-\cot(48)+...+\cot(133)-\cot(134)$. Observe that this "almost telescopes," if only we had some extra terms. Consider adding the sequence $\frac{1}{\sin(46)\sin(47)}+\frac{1}{\sin(48)\sin(49)}+...+\frac{1}{\sin(134)\sin(135)}$. By the identity $\sin(x)=\sin(180-x)$, this sequence is equal to the original one, simply written backwards. By the same logic as before, we may rewrite this sequence as $S\sin(1)=\cot(46)-\cot(47)+\cot(48)-\cot(49)+...+\cot(134)-\cot(135)$, and when we add the two sequences, they telescope to give $2S\sin(1)=\cot(45)-\cot(135)=2$. Hence, $S=\frac{1}{\sin(1^\circ)}$, and our angle is $\boxed{001}$.

~keeper1098

Solution 4

First, multiply $\sin n^{\circ}$ on both sides. \begin{align*} \frac{\sin n^\circ}{\sin m^\circ \sin (m+1)^\circ} &= \frac{\sin (k+n-k)^\circ}{\sin m^\circ \sin (m+1)^\circ} \\ &= \frac{\sin (k+n)^\circ \cos k^\circ - \sin k^\circ \cos (k+n)^\circ}{\sin m^\circ \sin (m+1)^\circ} \end{align*} Let $k = m$ since $k$ is could be any number. \begin{align*} &\quad \ \frac{\sin (k+n)^\circ \cos k^\circ}{\sin m^\circ \sin (m+1)^\circ} - \frac{\sin k^\circ \cos (k+n)^\circ}{\sin m^\circ \sin (m+1)^\circ} \\[0.5em] &=\frac{\sin (m+n)^\circ \cos m^\circ}{\sin m^\circ \sin (m+1)^\circ} - \frac{\sin m^\circ \cos (m+n)^\circ}{\sin m^\circ \sin (m+1)^\circ} \end{align*}

Lemma

   $n$ is equal to $1$.

Proof

\begin{align*} &=\frac{\sin (m+1)^\circ \cos m^\circ}{\sin m^\circ \sin (m+1)^\circ} - \frac{\sin m^\circ \cos (m+1)^\circ}{\sin m^\circ \sin (m+1)^\circ} \\[0.5em] &=\frac{\cos m^\circ}{\sin m^\circ} - \frac{\cos (m+1)^\circ}{\sin (m+1)^\circ} \\[0.5em] &= \cot m^\circ - \cot (m+1)^\circ \end{align*} The sum of all numbers could be written. Moreover, notice that $\cot\alpha + \cot\beta = 0$ if $\alpha + \beta = 180^\circ$. \begin{align*} &\quad \ \cot 45^\circ - \cot 46^\circ + \cot 47^\circ - \cot 48^\circ + \cdots - \cot 132^\circ + \cot 133^\circ - \cot 134^\circ \\ &= (\cot 45^\circ + \cot 47^\circ + \cdots + \cot 89^\circ + \cot 91^\circ + \dots + \cot 133^\circ) \\ &\qquad\qquad\qquad\qquad\qquad\qquad - (\cot 46^\circ + \dots + \cot 88^\circ + \cot 90^\circ + \cot 92^\circ + \dots + \cot 134^\circ) \\ &= \cot 45^\circ - \cot 90^\circ \\ &= 1 \end{align*} Because $1 = 1$, the lemma is true.

Q.E.D.

$n$ could be 1. Moreover, there are no smaller positive integer less than 1 to test. Thus, the least positive integer $n$ that satisfies the given condition is $\boxed{001}$.

~MaPhyCom

See also

2000 AIME II (Problems • Answer Key • Resources)
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