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 "2015 AIME I Problems/Problem 13"

(→‎Solution 3)
(→‎Solution 3)
Line 33: Line 33:
 
== Solution 3 ==
 
== Solution 3 ==
 
Similar to Solution <math>2</math>, so we use <math>\sin{2\theta}=2\sin\theta\cos\theta</math> and we find that:
 
Similar to Solution <math>2</math>, so we use <math>\sin{2\theta}=2\sin\theta\cos\theta</math> and we find that:
−
<cmath>\begin{align*}\sin4\sin8\sin12\sin16\cdots\sin84\sin88&=(2\sin2\cos2)(2\sin4\cos4)(2\sin6\cos6)(2\sin8\cos8)\cdots(2\sin42\cos42)(2\sin44\cos44)\\
+
<cmath>\begin{align*}\sin(4)\sin(8)\sin(12)\sin(16)\cdots\sin(84)\sin(88)&=(2\sin(2)\cos(2))(2\sin(4)\cos(4))(2\sin(6)\cos(6))(2\sin(8)\cos(8))\cdots(2\sin(42)\cos(42))(2\sin(44)\cos(44))\\
−
&=(2\sin2\sin88)(2\sin4\sin86)(2\sin6\sin84)(2\sin8\sin82)\cdots(2\sin42\sin48)(2\sin44\sin46)\\
+
&=(2\sin(2)\sin(88))(2\sin(4))\sin(86))(2\sin(6)\sin(84))(2\sin(8)\sin(82))\cdots(2\sin(42)\sin(48))(2\sin(44)\sin(46))\\
−
&=2^{22}(\sin2\sin88\sin4\sin86\sin6\sin84\sin8\sin82\cdots\sin42\sin48\sin44\sin46)\\
+
&=2^{22}(\sin(2)\sin(88)\sin(4)\sin(86)\sin(6)\sin(84)\sin(8)\sin(82)\cdots\sin(42)\sin(48)\sin(44)\sin(46))\\
−
&=2^{22}(\sin2\sin4\sin6\sin8\cdots\sin82\sin84\sin86\sin88)\end{align*}</cmath>
+
&=2^{22}(\sin(2)\sin(4)\sin(6)\sin(8)\cdots\sin(82)\sin(84)\sin(86)\sin(88))\end{align*}</cmath>
 
Now we can cancel the sines of the multiples of <math>4</math>:
 
Now we can cancel the sines of the multiples of <math>4</math>:
−
<cmath>1=2^{22}(\sin2\sin6\sin10\sin14\cdots\sin82\sin86)</cmath>
+
<cmath>1=2^{22}(\sin(2)\sin(6)\sin(10)\sin(14)\cdots\sin(82)\sin(86))</cmath>
−
So <math>\sin2\sin6\sin10\sin14\cdots\sin82\sin86=2^{-22}</math> and we can apply the double-angle formula again:
+
So <math>\sin(2)\sin(6)\sin(10)\sin(14)\cdots\sin(82)\sin(86)=2^{-22}</math> and we can apply the double-angle formula again:
−
<cmath>\begin{align*}2^{-22}&=\sin2\sin6\sin10\sin14\cdots\sin82\sin86\\
+
<cmath>\begin{align*}2^{-22}&=(\sin(2)\sin(6)\sin(10)\sin(14)\cdots\sin(82)\sin(86)\\
−
&=(2\sin1\cos1)(2\sin3\cos3)(2\sin5\cos5)(2\sin7\cos7)\cdots(2\sin41\cos41)(2\sin43\cos43)\\
+
&=(2\sin(1)\cos(1))(2\sin(3)\cos(3))(2\sin(5)\cos(5))(2\sin(7)\cos(7))\cdots(2\sin(41)\cos(41))(2\sin(43)\cos(43))\\
−
&=(2\sin1\sin89)(2\sin3\sin87)(2\sin5\sin85)(2\sin7\sin87)\cdots(2\sin41\sin49)(2\sin43\sin47)\\
+
&=(2\sin(1)\sin(89))(2\sin(3)\sin(87))(2\sin(5)\sin(85))(2\sin(7)\sin(87))\cdots(2\sin(41)\sin(49))(2\sin(43)\sin(47))\\
−
&=2^{22}(\sin1\sin89\sin3\sin87\sin5\sin85\sin7\sin83\cdots\sin41\sin49\sin43\sin47)\\
+
&=2^{22}(\sin(1)\sin(89)\sin(3)\sin(87)\sin(5)\sin(85)\sin(7)\sin(83)\cdots\sin(41)\sin(49)\sin(43)\sin(47))\\
−
&=2^{22}(\sin1\sin3\sin5\sin7\cdots\sin41\sin43)(\sin47\sin49\cdots\sin83\sin85\sin87\sin89)\end{align*}</cmath>
+
&=2^{22}(\sin(1)\sin(3)\sin(5)\sin(7)\cdots\sin(41)\sin(43))(\sin(47)\sin(49)\cdots\sin(83)\sin(85)\sin(87)\sin(89))\end{align*}</cmath>
−
Of course, <math>\sin45=2^{-\frac{1}{2}}</math> is missing, so we multiply it to both sides:
+
Of course, <math>\sin(45)=2^{-\frac{1}{2}}</math> is missing, so we multiply it to both sides:
−
<cmath>2^{-22}\sin45=2^{22}(\sin1\sin3\sin5\sin7\cdots\sin41\sin43)(\sin45)(\sin47\sin49\cdots\sin83\sin85\sin87\sin89)</cmath>
+
<cmath>2^{-22}\sin(45)=2^{22}(\sin(1)\sin(3)\sin(5)\sin(7)\cdots\sin(41)\sin(43))(\sin(45))(\sin(47)\sin(49)\cdots\sin(83)\sin(85)\sin(87)\sin(89))</cmath>
−
<cmath>(2^{-22})(2^{-\frac{1}{2}})=2^{22}(\sin1\sin3\sin5\sin7\cdots\sin83\sin85\sin87\sin89)</cmath>
+
<cmath>\left(2^{-22}\right)\left(2^{-\frac{1}{2}}\right)=2^{22}(\sin(1)\sin(3)\sin(5)\sin(7)\cdots\sin(83)\sin(85)\sin(87)\sin(89))</cmath>
−
<cmath>2^{-\frac{45}{2}}=2^{22}(\sin1\sin3\sin5\sin7\cdots\sin83\sin85\sin87\sin89)</cmath>
+
<cmath>2^{-\frac{45}{2}}=2^{22}(\sin(1)\sin(3)\sin(5)\sin(7)\cdots\sin(83)\sin(85)\sin(87)\sin(89))</cmath>
 
Now isolate the product of the sines:
 
Now isolate the product of the sines:
−
<cmath>\sin1\sin3\sin5\sin7\cdots\sin83\sin85\sin87\sin89=2^{-\frac{89}{2}}</cmath>
+
<cmath>\sin(1)\sin(3)\sin(5)\sin(7)\cdots\sin(83)\sin(85)\sin(87)\sin(89)=2^{-\frac{89}{2}}</cmath>
 
And the product of the squares of the cosecants as asked for by the problem is the square of the inverse of this number:
 
And the product of the squares of the cosecants as asked for by the problem is the square of the inverse of this number:
−
<cmath>\csc^21\csc^23\csc^25\csc^27\cdots\csc^283\csc^285\csc^287\csc^289=(\frac{1}{2^{\frac{89}{2}}})^2=(2^{\frac{89}{2}})^2=2^{89}</cmath>
+
<cmath>\csc^2(1)\csc^2(3)\csc^2(5)\csc^2(7)\cdots\csc^2(83)\csc^2(85)\csc^2(87)\csc^2(89)=\left(\frac{1}{2^{\frac{89}{2}}})^2=\left(2^{\frac{89}{2}})^2=2^{89}</cmath>
 
The answer is therefore <math>m+n=(2)+(89)=\boxed{091}</math>.
 
The answer is therefore <math>m+n=(2)+(89)=\boxed{091}</math>.
  

Revision as of 17:12, 21 March 2015

Problem

With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.

Solution 1

Let $x = \cos 1^\circ + i \sin 1^\circ$. Then from the identity \[\sin 1 = \frac{x - \frac{1}{x}}{2i} = \frac{x^2 - 1}{2 i x},\] we deduce that (taking absolute values and noticing $|x| = 1$) \[|2\sin 1| = |x^2 - 1|.\] But because $\csc$ is the reciprocal of $\sin$ and because $\sin z = \sin (180^\circ - z)$, if we let our product be $M$ then \[\frac{1}{M} = \sin 1^\circ \sin 3^\circ \sin 5^\circ \dots \sin 177^\circ \sin 179^\circ\] \[= \frac{1}{2^{90}} |x^2 - 1| |x^6 - 1| |x^{10} - 1| \dots |x^{354} - 1| |x^{358} - 1|\] because $\sin$ is positive in the first and second quadrants. Now, notice that $x^2, x^6, x^{10}, \dots, x^{358}$ are the roots of $z^{90} + 1 = 0.$ Hence, we can write $(z - x^2)(z - x^6)\dots (z - x^{358}) = z^{90} + 1$, and so \[\frac{1}{M} = \dfrac{1}{2^{90}}|1 - x^2| |1 - x^6| \dots |1 - x^{358}| = \dfrac{1}{2^{90}} |1^{90} + 1| = \dfrac{1}{2^{89}}.\] It is easy to see that $M = 2^{89}$ and that our answer is $2 + 89 = \boxed{91}$.

Solution 2

Let $p=\sin1\sin3\sin5...\sin89$

\[p=\sqrt{\sin1\sin3\sin5...\sin177\sin179}\]

\[=\sqrt{\frac{\sin1\sin2\sin3\sin4...\sin177\sin178\sin179}{\sin2\sin4\sin6\sin8...\sin176\sin178}}\]

\[=\sqrt{\frac{\sin1\sin2\sin3\sin4...\sin177\sin178\sin179}{(2\sin1\cos1)\cdot(2\sin2\cos2)\cdot(2\sin3\cos3)\cdot....\cdot(2\sin89\cos89)}}\]

\[=\sqrt{\frac{1}{2^{89}}\frac{\sin90\sin91\sin92\sin93...\sin177\sin178\sin179}{\cos1\cos2\cos3\cos4...\cos89}}\]

$=\sqrt{\frac{1}{2^{89}}}$ because of the identity $\sin(90+x)=\cos(x)$

we want $\frac{1}{p^2}=2^{89}$

Thus the answer is $2+89=091$

Solution 3

Similar to Solution $2$, so we use $\sin{2\theta}=2\sin\theta\cos\theta$ and we find that: \begin{align*}\sin(4)\sin(8)\sin(12)\sin(16)\cdots\sin(84)\sin(88)&=(2\sin(2)\cos(2))(2\sin(4)\cos(4))(2\sin(6)\cos(6))(2\sin(8)\cos(8))\cdots(2\sin(42)\cos(42))(2\sin(44)\cos(44))\\ &=(2\sin(2)\sin(88))(2\sin(4))\sin(86))(2\sin(6)\sin(84))(2\sin(8)\sin(82))\cdots(2\sin(42)\sin(48))(2\sin(44)\sin(46))\\ &=2^{22}(\sin(2)\sin(88)\sin(4)\sin(86)\sin(6)\sin(84)\sin(8)\sin(82)\cdots\sin(42)\sin(48)\sin(44)\sin(46))\\ &=2^{22}(\sin(2)\sin(4)\sin(6)\sin(8)\cdots\sin(82)\sin(84)\sin(86)\sin(88))\end{align*} Now we can cancel the sines of the multiples of $4$: \[1=2^{22}(\sin(2)\sin(6)\sin(10)\sin(14)\cdots\sin(82)\sin(86))\] So $\sin(2)\sin(6)\sin(10)\sin(14)\cdots\sin(82)\sin(86)=2^{-22}$ and we can apply the double-angle formula again: \begin{align*}2^{-22}&=(\sin(2)\sin(6)\sin(10)\sin(14)\cdots\sin(82)\sin(86)\\ &=(2\sin(1)\cos(1))(2\sin(3)\cos(3))(2\sin(5)\cos(5))(2\sin(7)\cos(7))\cdots(2\sin(41)\cos(41))(2\sin(43)\cos(43))\\ &=(2\sin(1)\sin(89))(2\sin(3)\sin(87))(2\sin(5)\sin(85))(2\sin(7)\sin(87))\cdots(2\sin(41)\sin(49))(2\sin(43)\sin(47))\\ &=2^{22}(\sin(1)\sin(89)\sin(3)\sin(87)\sin(5)\sin(85)\sin(7)\sin(83)\cdots\sin(41)\sin(49)\sin(43)\sin(47))\\ &=2^{22}(\sin(1)\sin(3)\sin(5)\sin(7)\cdots\sin(41)\sin(43))(\sin(47)\sin(49)\cdots\sin(83)\sin(85)\sin(87)\sin(89))\end{align*} Of course, $\sin(45)=2^{-\frac{1}{2}}$ is missing, so we multiply it to both sides: \[2^{-22}\sin(45)=2^{22}(\sin(1)\sin(3)\sin(5)\sin(7)\cdots\sin(41)\sin(43))(\sin(45))(\sin(47)\sin(49)\cdots\sin(83)\sin(85)\sin(87)\sin(89))\] \[\left(2^{-22}\right)\left(2^{-\frac{1}{2}}\right)=2^{22}(\sin(1)\sin(3)\sin(5)\sin(7)\cdots\sin(83)\sin(85)\sin(87)\sin(89))\] \[2^{-\frac{45}{2}}=2^{22}(\sin(1)\sin(3)\sin(5)\sin(7)\cdots\sin(83)\sin(85)\sin(87)\sin(89))\] Now isolate the product of the sines: \[\sin(1)\sin(3)\sin(5)\sin(7)\cdots\sin(83)\sin(85)\sin(87)\sin(89)=2^{-\frac{89}{2}}\] And the product of the squares of the cosecants as asked for by the problem is the square of the inverse of this number: 

\[\csc^2(1)\csc^2(3)\csc^2(5)\csc^2(7)\cdots\csc^2(83)\csc^2(85)\csc^2(87)\csc^2(89)=\left(\frac{1}{2^{\frac{89}{2}}})^2=\left(2^{\frac{89}{2}})^2=2^{89}\] (Error compiling LaTeX. Unknown error_msg)

The answer is therefore $m+n=(2)+(89)=\boxed{091}$.

See Also

2015 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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=2015_AIME_I_Problems/Problem_13&oldid=69369"