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

Revision as of 10:12, 14 December 2020

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, $(1+x)(1+y) = (1-x)(1-y)$ .

Notice here that we cancelled out one factor of (x-1) and (y-1), which implies that (x-1) and (y-1) were not 0. If indeed they were 0 though, we would have

$cos(3A)-1=0, cos(3A)=1$

For this we could say that A must be 120 degrees for this to work. This is one case. The B case follows in the same way, where B must be equal to 120 degrees. This doesn't change the overall solution though, as then the other angles are irrelevant (this is the largest angle, implying that this will have the longest side and so we would want to have the 120 degreee angle opposite of the unknown side).

Expanding, $1+x+y+xy=1-x-y+xy\rightarrow x+y=-x-y$ .

Simplification leads to $x+y=0$ .

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:

$m=10^2+13^2-2\cdot 10\cdot 13\cos\angle C$ $\rightarrow m=269-260\cos 120^\circ=269-260\left(\text{-}\frac{1}{2}\right)$ $\rightarrow m=269+130=399$

$\framebox{399}$

NOTE: This solution forgot the case of dividing by 0

NOTE FROM DIFFERENT PERSON: Pretty sure they addressed that in the notice

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 the Law of Cosines to get $\boxed{399}$

Note: This solution also forgets that $A$ or $B$ might be 120 when dividing by $\sin 3A$ and $\sin 3B$

Solution 3

$\cos3A+\cos3B=1-\cos(3C)=1+\cos(3A+3B)$ $2\cos\frac{3}{2}(A+B)\cos\frac{3}{2}(A-B)=2\cos^2\frac{3}{2}(A+B)$ If $\cos\frac{3}{2}(A+B) = 0$ , then $\frac{3}{2}(A+B)=90$ , $A+B=60$ , so $C=120$ ; otherwise, $2\cos\frac{3}{2}(A-B)=2cos\frac{3}{2}(A+B)$ $\sin\frac{3}{2}A\sin\frac{3}{2}B=0$ so either $\sin\frac{3}{2}A=0$ or $\sin\frac{3}{2}B=0$ , i.e., either $A=120$ or $B=120$ . In all cases, one of the angles must be 120, which opposes the longest side. Final result follows. $\boxed{399}$

-Mathdummy

Solution 4(quick sketch solve)

Let $BC$ be the unknown side length. By Law of Cosines we have that $BC = \sqrt{269-260\cos{A}}$ . We notice that $\cos{A}$ should be negative to optimize $BC$ so $A$ is between $90$ and $180$ degrees. We also know that the value inside the square root is an integer $m$ , so $269-260\cos{A}$ should be an integer. We can then assume that $A$ is $120$ degrees so $\cos{A} = \frac{-1}{2}$ . We do this because $120$ degrees is a "common" value and it makes the value inside the square root an integer. Plugging this into $269-260\cos{A}$ for $m$ we get that it is $\boxed{399}$ .

-srisainandan6

@@ Line 64: / Line 64: @@
 ==Solution 4(quick sketch solve)==
-Let <math>BC</math> be the unknown side length. By Law of Cosines we have that <math>BC = \sqrt{269-260\cos{A}}</math>. We notice that angle <math>A</math> should be negative to optimize <math>BC</math> so <math>A</math> is between <math>90</math> and <math>180</math> degrees. We also know that the value inside the square root is an integer <math>m</math>, so <math>269-260\cos{A}</math> should be an integer. We can then assume that <math>A</math> is <math>120</math> degrees so <math>\cos{A} = \frac{-1}{2}</math>. We do this because <math>120</math> degrees is a "common" value and it makes the value inside the square root an integer. Plugging this into <math>269-260\cos{A}</math> for <math>m</math> we get that it is <math>\boxed{399}</math>.
+Let <math>BC</math> be the unknown side length. By Law of Cosines we have that <math>BC = \sqrt{269-260\cos{A}}</math>. We notice that <math>\cos{A}</math> should be negative to optimize <math>BC</math> so <math>A</math> is between <math>90</math> and <math>180</math> degrees. We also know that the value inside the square root is an integer <math>m</math>, so <math>269-260\cos{A}</math> should be an integer. We can then assume that <math>A</math> is <math>120</math> degrees so <math>\cos{A} = \frac{-1}{2}</math>. We do this because <math>120</math> degrees is a "common" value and it makes the value inside the square root an integer. Plugging this into <math>269-260\cos{A}</math> for <math>m</math> we get that it is <math>\boxed{399}</math>.
 -srisainandan6

2014 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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