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Difference between revisions of "1995 AIME Problems/Problem 7"

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where <math>k, m,</math> and <math>n_{}</math> are [[positive integer]]s with <math>m_{}</math> and <math>n_{}</math> [[relatively prime]], find <math>k+m+n.</math>
 
where <math>k, m,</math> and <math>n_{}</math> are [[positive integer]]s with <math>m_{}</math> and <math>n_{}</math> [[relatively prime]], find <math>k+m+n.</math>
  
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== Solution ==
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== Solution 1 ==
 
From the givens,  
 
From the givens,  
 
<math>2\sin t \cos t + 2 \sin t + 2 \cos t = \frac{1}{2}</math>, and adding <math>\sin^2 t + \cos^2t = 1</math> to both sides gives <math>(\sin t + \cos t)^2 + 2(\sin t + \cos t) = \frac{3}{2}</math>.  Completing the square on the left in the variable <math>(\sin t + \cos t)</math> gives <math>\sin t + \cos t = -1 \pm \sqrt{\frac{5}{2}}</math>.  Since <math>|\sin t + \cos t| \leq \sqrt 2 < 1 + \sqrt{\frac{5}{2}}</math>, we have <math>\sin t + \cos t = \sqrt{\frac{5}{2}} - 1</math>.  Subtracting twice this from our original equation gives <math>(\sin t - 1)(\cos t - 1) = \sin t \cos t - \sin t - \cos t + 1 = \frac{13}{4} - \sqrt{10}</math>, so the answer is <math>13 + 4 + 10 = \boxed{027}</math>.
 
<math>2\sin t \cos t + 2 \sin t + 2 \cos t = \frac{1}{2}</math>, and adding <math>\sin^2 t + \cos^2t = 1</math> to both sides gives <math>(\sin t + \cos t)^2 + 2(\sin t + \cos t) = \frac{3}{2}</math>.  Completing the square on the left in the variable <math>(\sin t + \cos t)</math> gives <math>\sin t + \cos t = -1 \pm \sqrt{\frac{5}{2}}</math>.  Since <math>|\sin t + \cos t| \leq \sqrt 2 < 1 + \sqrt{\frac{5}{2}}</math>, we have <math>\sin t + \cos t = \sqrt{\frac{5}{2}} - 1</math>.  Subtracting twice this from our original equation gives <math>(\sin t - 1)(\cos t - 1) = \sin t \cos t - \sin t - \cos t + 1 = \frac{13}{4} - \sqrt{10}</math>, so the answer is <math>13 + 4 + 10 = \boxed{027}</math>.
  
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==Solution 2==
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== Solution 2 ==
−
Let <math>(1 - \sin t)(1 - \cos t) = x</math>. Multiplying <math>x</math> with the given equation, <math>\frac{5x}{4} = (1 - \sin^2 t)(1 - \cos^2 t) = \sin^2 t \cos ^2 t</math>, and <math>\frac{\sqrt{5x}}{2} = \sin t \cos t</math>. Simplifying and rearranging the given equation, <math>\sin t + \cos t = \frac{5}{4} - (\sin^2 t + \cos^2 t) - \sin t \cos t = \frac{1}{4} - \frac{\sqrt{5x}}{2}</math>. Notice that <math>(1 + \sin t)(1 + \cos t) - 2(\sin t + \cos t) = x</math>, and substituting, <math>x = \frac{5}{4} - 2( \frac{1}{4} - \frac{\sqrt{5x}}{2}) = \frac{3}{4} + \sqrt{5x}</math>. Rearranging, <math>\sqrt{5x} = x - \frac{3}{4}</math>, and squaring, <math>5x = x^2 - \frac{3}{2} x + \frac{9}{16}</math>, so <math>x^2 - \frac{13}{2} + \frac{9}{16} = 0</math>, and <math>x = \frac{13}{4} \pm \sqrt{10}</math>, but as both parts of the product <math>x</math> doesn't exceed <math>2</math>, <math>x < 4</math>. Therefore, <math>x = \frac{13}{4} - \sqrt{10}</math>, and the answer is <math> 13 + 4 + 10 = \boxed{027}</math>.
+
Let <math>(1 - \sin t)(1 - \cos t) = x</math>. Multiplying <math>x</math> with the given equation, <math>\frac{5x}{4} = (1 - \sin^2 t)(1 - \cos^2 t) = \sin^2 t \cos ^2 t</math>, and <math>\frac{\sqrt{5x}}{2} = \sin t \cos t</math>. Simplifying and rearranging the given equation, <math>\sin t + \cos t = \frac{5}{4} - (\sin^2 t + \cos^2 t) - \sin t \cos t = \frac{1}{4} - \frac{\sqrt{5x}}{2}</math>. Notice that <math>(1 + \sin t)(1 + \cos t) - 2(\sin t + \cos t) = x</math>, and substituting, <math>x = \frac{5}{4} - 2( \frac{1}{4} - \frac{\sqrt{5x}}{2}) = \frac{3}{4} + \sqrt{5x}</math>. Rearranging and squaring, <math>5x = x^2 - \frac{3}{2} x + \frac{9}{16}</math>, so <math>x^2 - \frac{13}{2} x + \frac{9}{16} = 0</math>, and <math>x = \frac{13}{4} \pm \sqrt{10}</math>, but clearly, <math>0 \leq x < 4</math>. Therefore, <math>x = \frac{13}{4} - \sqrt{10}</math>, and the answer is <math> 13 + 4 + 10 = \boxed{027}</math>.
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== Solution 3 ==
 +
 
 +
We want <math>1+\sin t \cos t-\sin t-\cos t</math>. However, note that we only need to find <math>\sin t+\cos t</math>.
 +
 
 +
Let <math>y = \sin t+\cos t \rightarrow y^2 = \sin^2 t + \cos^2 t + 2\sin t \cos t = 1 + 2\sin t \cos t</math>
 +
 
 +
From this we have <math>\sin t \cos t = \frac{y^2-1}{2}</math> and <math>\sin t + \cos t = y</math>
 +
 
 +
Substituting, we have <math>2y^2+4y-3=0 \rightarrow y = \frac {-2 \pm \sqrt{10}}{2}</math>
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<math>\frac{5}{4} - 2(\frac{-2+\sqrt{10}}{2}) = \frac{13}{4}-\sqrt{10} \rightarrow 13+10+4=\boxed{027}</math>.
  
 
== See also ==
 
== See also ==

Latest revision as of 08:23, 13 December 2023

Problem

Given that $(1+\sin t)(1+\cos t)=5/4$ and

$(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},$

where $k, m,$ and $n_{}$ are positive integers with $m_{}$ and $n_{}$ relatively prime, find $k+m+n.$

Solution 1

From the givens, $2\sin t \cos t + 2 \sin t + 2 \cos t = \frac{1}{2}$, and adding $\sin^2 t + \cos^2t = 1$ to both sides gives $(\sin t + \cos t)^2 + 2(\sin t + \cos t) = \frac{3}{2}$. Completing the square on the left in the variable $(\sin t + \cos t)$ gives $\sin t + \cos t = -1 \pm \sqrt{\frac{5}{2}}$. Since $|\sin t + \cos t| \leq \sqrt 2 < 1 + \sqrt{\frac{5}{2}}$, we have $\sin t + \cos t = \sqrt{\frac{5}{2}} - 1$. Subtracting twice this from our original equation gives $(\sin t - 1)(\cos t - 1) = \sin t \cos t - \sin t - \cos t + 1 = \frac{13}{4} - \sqrt{10}$, so the answer is $13 + 4 + 10 = \boxed{027}$.

Solution 2

Let $(1 - \sin t)(1 - \cos t) = x$. Multiplying $x$ with the given equation, $\frac{5x}{4} = (1 - \sin^2 t)(1 - \cos^2 t) = \sin^2 t \cos ^2 t$, and $\frac{\sqrt{5x}}{2} = \sin t \cos t$. Simplifying and rearranging the given equation, $\sin t + \cos t = \frac{5}{4} - (\sin^2 t + \cos^2 t) - \sin t \cos t = \frac{1}{4} - \frac{\sqrt{5x}}{2}$. Notice that $(1 + \sin t)(1 + \cos t) - 2(\sin t + \cos t) = x$, and substituting, $x = \frac{5}{4} - 2( \frac{1}{4} - \frac{\sqrt{5x}}{2}) = \frac{3}{4} + \sqrt{5x}$. Rearranging and squaring, $5x = x^2 - \frac{3}{2} x + \frac{9}{16}$, so $x^2 - \frac{13}{2} x + \frac{9}{16} = 0$, and $x = \frac{13}{4} \pm \sqrt{10}$, but clearly, $0 \leq x < 4$. Therefore, $x = \frac{13}{4} - \sqrt{10}$, and the answer is $13 + 4 + 10 = \boxed{027}$.

Solution 3

We want $1+\sin t \cos t-\sin t-\cos t$. However, note that we only need to find $\sin t+\cos t$.

Let $y = \sin t+\cos t \rightarrow y^2 = \sin^2 t + \cos^2 t + 2\sin t \cos t = 1 + 2\sin t \cos t$

From this we have $\sin t \cos t = \frac{y^2-1}{2}$ and $\sin t + \cos t = y$

Substituting, we have $2y^2+4y-3=0 \rightarrow y = \frac {-2 \pm \sqrt{10}}{2}$

$\frac{5}{4} - 2(\frac{-2+\sqrt{10}}{2}) = \frac{13}{4}-\sqrt{10} \rightarrow 13+10+4=\boxed{027}$.

See also

1995 AIME (Problems • Answer Key • Resources)
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