Difference between revisions of "2012 AIME I Problems/Problem 13"

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Problem 13

Three concentric circles have radii $3,$ $4,$ and $5.$ An equilateral triangle with one vertex on each circle has side length $s.$ The largest possible area of the triangle can be written as $a + \tfrac{b}{c} \sqrt{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisible by the square of any prime. Find $a+b+c+d.$

Solution

Solution 1

Reinterpret the problem in the following manner. Equilateral triangle $ABC$ has a point $X$ on the interior such that $AX = 5,$ $BX = 4,$ and $CX = 3.$ A $60^\circ$ counter-clockwise rotation about vertex $A$ maps $X$ to $X'$ and $B$ to $C.$ Note that angle $XAX'$ is $60$ and $XA = X'A = 5$ which tells us that triangle $XAX'$ is equilateral and that $XX' = 5.$ We now notice that $XC = 3$ and $X'C = 4$ which tells us that angle $XCX'$ is $90$ because there is a $3$ - $4$ - $5$ Pythagorean triple. Now note that $\angle ABC + \angle ACB = 120^\circ$ and $\angle XCA + \angle XBA = 90^\circ,$ so $\angle XCB+\angle XBC = 30^\circ$ and $\angle BXC = 150^\circ.$ Applying the law of cosines on triangle $BXC$ yields

$BC^2 = BX^2+CX^2 - 2 \cdot BX \cdot CX \cdot \cos150^\circ = 4^2+3^2-24 \cdot \frac{-\sqrt{3}}{2} = 25+12\sqrt{3}$

and thus the area of $ABC$ equals $BC^2\frac{\sqrt{3}}{4} = 25\frac{\sqrt{3}}{4}+9.$

so our final answer is $3+4+25+9 = \boxed{041}.$

Remark: The new figure (the rotations and the triangle) must be twice the original triangle's area. So it is simply $\frac{\frac{9\sqrt3+16\sqrt3+25\sqrt3}{4}+3(\frac{3*4}{2})}{2}=25\frac{\sqrt{3}}{4}+9$

Solution 2

Here is a proof that shows that there are 2 distinct equilateral triangles (up to congruence) that have the given properties: We claim that there are 2 distinct equilateral triangles (up to congruence) that have the given properties; one of which has largest area. We have 2 cases to consider; either the center $O$ of the circles lies in the interior of triangle $ABC$ or it does not (and we shall show that both can happen). To see that the first case can occur, refer to Solution 1 above, or for a less creative and more direct approach proceed as follows. Using the notation from Solution 1, let $\theta$ be the measure of angle $XAC$ so that angle $BAX$ has measure $60-\theta$ . Let $AB=BC=AC=x$ . The law of cosines on triangles $BAX$ and $XAC$ yields $\cos(60-\theta)=\frac{x^2+9}{10x}$ and $\cos\theta=\frac{x^2+16}{10x}$ . Solving this system will yield the value of $x$ . Since $\cos\theta=\frac{x^2+16}{10x}$ we have that $\sin\theta=\frac{\sqrt{100x^2-(x^2+16)^2}}{10x}$ . Substituting these into the equation $\frac{x^2+9}{10x}=\cos(60-\theta)=\frac{1}{2}\cos\theta+\frac{\sqrt{3}}{2}\sin\theta$ we obtain $\frac{x^2+9}{10x}=\frac{1}{2}\frac{x^2+16}{10x}+\frac{\sqrt{3}}{2}\frac{\sqrt{100x^2-(x^2+16)^2}}{10x}$ . After clearing denominators, combining like terms, isolating the square root, squaring, and expanding, we obtain $x^4-50x^2+193=0$ so that by the quadratic formula $x^2=25\pm12\sqrt{3}$ . Under the hypothesis that $O$ lies in the interior of triangle $ABC$ , $x^2$ must be $25+12\sqrt{3}$ . To see this, note that the other value for $x^2$ is roughly $4.2$ so that $x\approx 2.05$ , but since $AX=5$ and $AX\leq x$ we have a contradiction. We then obtain the area as in Solution 1.

Now, suppose $O$ does not lie in the interior of triangle $ABC$ . We then obtain convex quadrilateral $OBAC$ with diagonals $CB$ and $OA$ intersecting at $X$ . Here $AX=AB=AC=x$ . We may let $\alpha$ denote the measure of angle $CAX$ so that angle $XAB$ measures $60-\alpha$ . Note that the law of cosines on triangles $CXA$ and $BXA$ yield the same equations as in the first case with $\theta$ replaced with $\alpha$ . Thus we obtain again $x^2=25\pm12\sqrt{3}$ . If $x^2=25+12\sqrt{3}$ then $x\approx 6.8$ , but this is impossible since $AX\leq 5$ but the shortest possible distance from $A$ to $X$ is the height of equilateral triangle $ABC$ which is $\approx6.8\sqrt{3}\approx5.8$ ; a contradiction. Hence in this case $x^2=25-12\sqrt{3}$ . But, the area of this triangle is clearly less than that in the first case, so we are done. Hence the phrasing of the question (the triangle with maximal area) is absolutely necessary since there are 2 possible triangles (up to congruence).

Solution 3

The problem basically asks for the area of $\bigtriangleup ABC$ such that it is equilateral and there is a point $O$ inside of the triangle which satisfies $AO = 3$ , $BO = 4$ , and $CO = 5$ .

Let $AB = BC = AC = s$ . We want the area of the triangle, which is just $[ABC] = \frac{s^{2}\sqrt{3}}{4}$ . Thus, we want to know $s^{2}$ , and then finding the area will be a matter of simple calculation.

By law of cosines, $s^{2} = 3^{2} + 4^{2} - 2 \cdot 3 \cdot 4 \cdot \cos{\theta} = 25 - 24\cos{\theta}$ , where $\theta = \angle{AOB}$ .

Now, if we plug this value in for $s^{2}$ in the area formula, we get $[ABC] = \frac{25}{4}\sqrt{3} - 6\sqrt{3}\cos{\theta}$ .

Notice that, in order for this expression to be in the required answer form $a + \tfrac{b}{c} \sqrt{d}$ , $\cos{\theta}$ must involve $\sqrt{3}$ . Now, even minimal experience with simple trigonometric functions will instantly make you think of $\frac{\pi}{6}$ or $\frac{5\pi}{6}$ . The former doesn't work, since $\angle{AOB}$ is obtuse, so $\theta = \frac{5\pi}{6}$ .

Thus, our area must be $[ABC] = \frac{25}{4}\sqrt{3} + 9$ , and so our answer is $\boxed{041}$ .

$\textbf{NOTE:}$

When we were using law of cosines, we instantly went to the sub-triangle $\bigtriangleup ABO$ . Why?

Notice that $\angle{ABO}$ is bigger than both $\angle{BOC}$ and $\angle{AOC}$ because the sides which meet to form the angle are shorter for $\angle{ABO}$ than the other two triangles.

Also notice that the three angles add up to $2\pi$ radians. One of our angles is $\frac{5\pi}{6}$ by our reasoning/guessing above. This angle then must be the greatest angle out of the three, since the three angles all must be obtuse, and if one angle was $\frac{5\pi}{6}$ , and another was greater than $\frac{5\pi}{6}$ , then the third angle would be less than $\frac{\pi}{3}$ , leading to a contradiction, since we know that all of the three angles are obtuse.

Thus, we know that our greatest angle $\angle{ABO} = \frac{5\pi}{6}$ . So, we instantly went to the sub-triangle $\bigtriangleup ABO$ when using law of cosines since the other sub-triangles will get us nowhere.

Solution 4

$[asy] import olympiad; import cse5; import graph; dotfactor = 2; unitsize(0.3inch); pair B = (0,0), C= (5,0), A = (sqrt(9-2.4*2.4),2.4); pair D = rotate(60,B)*A, E=rotate(60,A)*C, F=rotate(60,C)*B; pair X = extension(A,F,D,C); pair L = (-1.5,2), M = (6.2,3), N = rotate(-60,L)*M; dot("$C$", C, dir(0)); dot("$A$", A, dir(90));dot("$B$", B, dir(180)); dot("$D$", D, NE); dot("$E$", E, dir(90));dot("$F$", F, dir(270)); dot("$M$", M, NE); dot("$N$", N, dir(270));dot("$L$", L, NW); dot("$X$", X, dir(250)); draw(L--X); draw(M--X); draw(N--X); draw(A--B--C--cycle); draw(A--D--B); draw(B--F--C); draw(A--E--C); draw(A--F,dashed); draw(D--C,dashed); draw(B--E,dashed); draw(L--M--N--cycle); [/asy]$

Let's call the circle center $X$ . It has a distance of 3, 4, 5 to an equilateral triangle $LMN$ . Consider $X$ ’s pedal triangle $ABC$ . Since $X$ ’s antipedal triangle is equilateral, $X$ must be the one of the isogonic centers of $\triangle{ABC}$ . We’ll take the one inside $ABC$ , i.e., the Fermat point, because it leads to larger $\triangle LMN$ . Now we construct the three equilateral triangles $ABD$ , $ACE$ , and $BCF$ , the same way the Fermat point is constructed. Then we have $\angle DXE = \angle EXF = \angle FXE = 120$ . Since $AEMCX$ is concyclic with $XM$ =4 as diameter, we have $AC=4\sin(60)$ . Similarly, $AB=3\sin(60)$ , and $BC=5\sin(60)$ . So $\triangle ABC$ is a 3-4-5 right triangle with $\angle BAC=90$ . With some more angle chasing we get $\angle MXC+\angle LXB = \angle MAC + \angle LAB = 180 – \angle BAC = 90$ $\angle LXM = 360 – (\angle MXC + \angle LXB + \angle BXC) = 360 –(90+120)=150$ By Law of Cosines, we have $LM^2 = 3^2+4^2-2*3*4\cos(150)=25+12\sqrt 3$ And the area follows. $[LMN] = \frac{25}{4}\sqrt{3} + 9; \boxed{041}.$ By Mathdummy

Solution 5

Let $ABC$ be the equilateral triangle with $AB=BC=CA=x.$ Assume the coordinates of the vertices are $A(-\dfrac{x}{2},0), B(\dfrac{x}{2},0)$ and $C(0,\dfrac{\sqrt{3}}{2}x).$ Let $P(a,b)$ be such that $PA=3, PB=4$ and $PC=5.$ Then $\left (a+\dfrac{x}{2}\right )^2 +b^2=3^2,$ $\left (a-\dfrac{x}{2}\right )^2 +b^2=4^2,$ $a^2+\left (b-\dfrac{\sqrt{3}}{2}x\right )^2 =5^2.$ Subtraction and addition of the first two equations yield $2ax=-7, 2a^2+\dfrac{1}{2}x^2+2b^2=25.$ The third equation gives $2a^2+2b^2-2\sqrt{3}bx+\dfrac{3}{2}x^2=50.$ Then $x^2-2\sqrt{3}bx=25.$ We can then solve for $a, b$ in terms of $x$ and have a substitution. We have $2\left (\dfrac{-7}{2x}\right )^2 + \dfrac{1}{2}x^2 + 2\left (\dfrac{x^2-25}{2\sqrt{3}x}\right )^2 =25.$ Simplify it we have a quadratic equation for $x^2: x^4-50x^2+193=0.$ So $x^2=25\pm 12\sqrt{3}.$ The larger one leads to the solution. The smaller one relates to another equilateral triangle, as indicated in Solution 2.

-JZ

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2012aimei/353

~ dolphin7

@@ Line 98: / Line 98: @@
 ==See also==
-[[2020 AMC12A Problem 24|2020 AMC 12A Problems/Problem 24]]
+[[2020 AMC12A Problems/Problem 24|2020 AMC 12A Problems/Problem 24]]
 {{AIME box|year=2012|n=I|num-b=12|num-a=14}}
 {{MAA Notice}}

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