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Difference between revisions of "2014 AIME II Problems/Problem 12"

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== Solution 1 ==
 
== Solution 1 ==
−
Note that <math>\cos{3C}=-\cos{(3A+3B)}</math>. Thus, our expression is of the form <math>\cos{3A}+\cos{3B}-\cos{(3A+3B)}=1</math>. Let <math>\cos{3A}=x</math> and <math>\cos{3B}=y</math>. Expanding, we get <math>x+y-xy+\sqrt{1-x^2}\sqrt{1-y^2}=1</math>, or <math>\sqrt{1-x^2}\sqrt{1-y^2}=(x-1)(y-1)</math>.  
+
Note that <math>\cos{3C}=-\cos{(3A+3B)}</math>. Thus, our expression is of the form <math>\cos{3A}+\cos{3B}-\cos{(3A+3B)}=1</math>. Let <math>\cos{3A}=x</math> and <math>\cos{3B}=y</math>. Using the fact that <math>\cos(3A+3B)=\cos 3A\cos 3B-\sin 3A\sin 3B=xy-\sqrt{1-x^2}\sqrt{1-y^2}</math>, we get <math>x+y-xy+\sqrt{1-x^2}\sqrt{1-y^2}=1</math>, or <math>\sqrt{1-x^2}\sqrt{1-y^2}=xy-x-y-1=(x-1)(y-1)</math>.  
  
 
Squaring both sides, we get <math>(1-x^2)(1-y^2) = [(x-1)(y-1)]^2</math>. Cancelling factors, <math>(1+x)(1+y) = (1-x)(1-y).</math>
 
Squaring both sides, we get <math>(1-x^2)(1-y^2) = [(x-1)(y-1)]^2</math>. Cancelling factors, <math>(1+x)(1+y) = (1-x)(1-y).</math>
−
Expanding, <math>xy+x+y+1 = 1-x-y+xy</math>.  Simplification leads to <math>x=-y</math>.
+
Expanding, <math>1+x+y+xy= 1-x-y+xy\rightarrow x+y=-x-y\rightarrow 2x=-2y</math>.  Simplification leads to <math>x=-y</math> and <math>x+y=0</math>.
  
−
Therefore, cos(3C) must equal 1. So C could be 0 or 120 degrees. We eliminate 0 and use law of cosines to get our answer: <math>\framebox{399}</math>
+
Therefore, <math>cos(3C)=1</math>. So <math>\angle C</math> could be <math>0^\circ</math> or <math>120^\circ</math>. We eliminate <math>0^\circ</math> and use law of cosines to get our answer:  
 +
<cmath>m=10^2+13^2-2\cdot 10\cdot 13\cos\angle C</cmath>
 +
<cmath>\rightarrow m=269-260\cos 120^\circ=269-260\left(\frac{1}{2}\right)</cmath>
 +
<cmath>\rightarrow m=269+130=399</cmath>
 +
 
 +
<math>\framebox{399}</math>
  
 
==Solution 2==
 
==Solution 2==

Revision as of 15:02, 9 February 2016

Problem

Suppose that the angles of $\triangle ABC$ satisfy $\cos(3A)+\cos(3B)+\cos(3C)=1.$ Two sides of the triangle have lengths 10 and 13. There is a positive integer $m$ so that the maximum possible length for the remaining side of $\triangle ABC$ is $\sqrt{m}.$ Find $m.$

Solution 1

Note that $\cos{3C}=-\cos{(3A+3B)}$. Thus, our expression is of the form $\cos{3A}+\cos{3B}-\cos{(3A+3B)}=1$. Let $\cos{3A}=x$ and $\cos{3B}=y$. Using the fact that $\cos(3A+3B)=\cos 3A\cos 3B-\sin 3A\sin 3B=xy-\sqrt{1-x^2}\sqrt{1-y^2}$, we get $x+y-xy+\sqrt{1-x^2}\sqrt{1-y^2}=1$, or $\sqrt{1-x^2}\sqrt{1-y^2}=xy-x-y-1=(x-1)(y-1)$.

Squaring both sides, we get $(1-x^2)(1-y^2) = [(x-1)(y-1)]^2$. Cancelling factors, $(1+x)(1+y) = (1-x)(1-y).$ Expanding, $1+x+y+xy= 1-x-y+xy\rightarrow x+y=-x-y\rightarrow 2x=-2y$. Simplification leads to $x=-y$ and $x+y=0$.

Therefore, $cos(3C)=1$. So $\angle C$ could be $0^\circ$ or $120^\circ$. We eliminate $0^\circ$ and use law of cosines to get our answer: \[m=10^2+13^2-2\cdot 10\cdot 13\cos\angle C\] \[\rightarrow m=269-260\cos 120^\circ=269-260\left(\frac{1}{2}\right)\] \[\rightarrow m=269+130=399\]

$\framebox{399}$

Solution 2

As above, we can see that $\cos3A+\cos3B-\cos(3A+3B)=1$

Expanding, we get

$\cos3A+\cos3B-\cos3A\cos3B+\sin3A\sin3B=1$

$\cos3A\cos3B-\cos3A-\cos3B+1=\sin3A\sin3B$

$(\cos3A-1)(\cos3B-1)=\sin3A\sin3B$

$\frac{\cos3A-1}{\sin3A}\cdot\frac{\cos3B-1}{\sin3B}=1$

$\tan{\frac{3A}{2}}\tan{\frac{3B}{2}}=1$

Note that $\tan{x}=\frac{1}{\tan(90-x)}$, or $\tan{x}\tan(90-x)=1$

Thus $\frac{3A}{2}+\frac{3B}{2}=90$, or $A+B=60$.

Now we know that $C=120$, so we can just use Law or Cosines to get $\boxed{399}$

See also

2014 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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