2024 AMC 12A Problems/Problem 18: Difference between revisions

Revision as of 14:23, 9 November 2024

Problem

On top of a rectangular card with sides of length $1$ and $2+\sqrt{3}$ , an identical card is placed so that two of their diagonals line up, as shown ( $\overline{AC}$ , in this case).

$[asy] defaultpen(fontsize(12)+0.85); size(150); real h=2.25; pair C=origin,B=(0,h),A=(1,h),D=(1,0),Dp=reflect(A,C)*D,Bp=reflect(A,C)*B; pair L=extension(A,Dp,B,C),R=extension(Bp,C,A,D); draw(L--B--A--Dp--C--Bp--A); draw(C--D--R); draw(L--C^^R--A,dashed+0.6); draw(A--C,black+0.6); dot("$C$",C,2*dir(C-R)); dot("$A$",A,1.5*dir(A-L)); dot("$B$",B,dir(B-R)); [/asy]$

Continue the process, adding a third card to the second, and so on, lining up successive diagonals after rotating clockwise. In total, how many cards must be used until a vertex of a new card lands exactly on the vertex labeled $B$ in the figure?

$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }\text{No new vertex will land on }B.$

Solution 1

Let the midpoint of $AC$ be $P$ .

We see that no matter how many moves we do, $P$ stays where it is.

Now we can find the angle of rotation ( $\angle APB$ ) per move with the following steps:

$AP^2=(\frac{1}{2})^2+(1+\frac{\sqrt{3}}{2})^2=2+\sqrt{3}$ $1^2=AP^2+AP^2-2(AP)(AP)\cos\angle APB$ $1=2(2+\sqrt{3})(1-\cos\angle APB)$ $\cos\angle APB=\frac{3+2\sqrt{3}}{4+2\sqrt{3}}$ $\cos\angle APB=\frac{3+2\sqrt{3}}{4+2\sqrt{3}}\cdot\frac{4-2\sqrt{3}}{4-2\sqrt{3}}$ $\cos\angle APB=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}$ $\angle APB=30^\circ$ Since Vertex $C$ is the closest one and $\angle BPC=360-180-30=150$

Vertex C will land on Vertex B when $\frac{150}{30}+1=\fbox{(A) 6}$ cards are placed.

(someone insert diagram maybe)

~lptoggled, minor Latex edits by eevee9406

Solution 2

AC intersects BD at O,

we want to find $\angle AOB$

since $tan(75^\circ) = 2+ \sqrt{3} = AD/AB$ , $\angle CBD = \angle BCA = 15^\circ$

so each time we rotate BD to AC for $30^\circ$ , and we need to rotate n = $180^\circ / 30^\circ = 6$ times to overlap with B (from one of A,B,C,D) ( should not be n = $360^\circ / 30^\circ = 12)$ $answer {(A) 6}$

note: if you don't remember $tan(75^\circ)$ $tan(75^\circ) = \frac{tan(45^\circ) + tan(30^\circ)}{ 1 - tan(45^\circ)*tan(30^\circ)}$ $= \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1* \frac{1}{\sqrt{3}}}$ $= \frac{(\sqrt{3}+1)^2 }{ (\sqrt{3})^2-1} = 2+ \sqrt{3}$

~luckuso

Solution 3(In case you have no time and that's what I did)

tan 15=sin15/cos15=1/(2+sqrt3) and it eliminates all options except 6 and 12. After one rotation it has turned 30degrees, so to satisfy the problem, divide 180 by 30 and you get 6

@@ Line 32: / Line 32: @@
 ==Solution 2==
+[[Image:2024_amc12A_p18.png|thumb|center|600px|]]
+AC intersects BD at O,
-AC intersect BD at O,
+we want to find <math>\angle AOB </math>
-since <math>cot(15^\circ) = 2+ \sqrt{3} </math>,  <math>\angle CBD = \angle BCA = 15^\circ </math>
+since  <math>tan(75^\circ) = 2+ \sqrt{3} = AD/AB </math>,  <math>\angle CBD = \angle BCA = 15^\circ </math>
   <cmath>\angle AOB  = \angle CBD  + \angle BCA  =30^\circ , </cmath>
-so each time rotate BD to AC for <math>30^\circ </math>
+so each time we rotate BD to AC for <math>30^\circ </math>,
-and we need n = <math> 180^\circ / 30^\circ = 6 </math>
+and we need to rotate n = <math> 180^\circ / 30^\circ = 6 </math> times to overlap with B (from one of  A,B,C,D)
 ( should not be n = <math>360^\circ / 30^\circ = 12) </math>
 <cmath>answer {(A) 6} </cmath>
+note: if you don't remember  <math>tan(75^\circ)</math>
+<cmath> tan(75^\circ) = \frac{tan(45^\circ) + tan(30^\circ)}{ 1 - tan(45^\circ)*tan(30^\circ)} </cmath>
+<cmath> = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1* \frac{1}{\sqrt{3}}} </cmath>
+<cmath> = \frac{(\sqrt{3}+1)^2  }{ (\sqrt{3})^2-1} = 2+ \sqrt{3} </cmath>
 ~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso]

2024 AMC 12A (Problems • Answer Key • Resources)
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