2010 AIME II Problems/Problem 9

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Problem 9

Let $ABCDEF$ be a regular hexagon. Let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $AF$, respectively. The segments $\overbar{AH}$ (Error compiling LaTeX. Unknown error_msg), $\overbar{BI}$ (Error compiling LaTeX. Unknown error_msg), $\overbar{CJ}$ (Error compiling LaTeX. Unknown error_msg), $\overbar{DK}$ (Error compiling LaTeX. Unknown error_msg), $\overbar{EL}$ (Error compiling LaTeX. Unknown error_msg), and $\overbar{FG}$ (Error compiling LaTeX. Unknown error_msg) bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ABCDEF$ be expressed as a fraction $\frac {m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

[asy] defaultpen(0.8pt+fontsize(12pt)); pair A,B,C,D,E,F; pair G,H,I,J,K,L; A=dir(0); B=dir(60); C=dir(120); D=dir(180); E=dir(240); F=dir(300); draw(A--B--C--D--E--F--cycle,blue);  G=(A+B)/2; H=(B+C)/2; I=(C+D)/2; J=(D+E)/2; K=(E+F)/2; L=(F+A)/2;  int i; for (i=0; i<6; i+=1) {  draw(rotate(60*i)*(A--H),dotted);  }   pair M,N,O,P,Q,R; M=extension(A,H,B,I); N=extension(B,I,C,J); O=extension(C,J,D,K); P=extension(D,K,E,L); Q=extension(E,L,F,G); R=extension(F,G,A,H); draw(M--N--O--P--Q--R--cycle,red);   label('A',A, E); label('B',B,NE); label('C',C,NW); label('D',D, W); label('E',E,SW); label('F',F,SE); label('G',G,NE); label('H',H, N); label('I',I,NW); label('J',J,SW); label('K',K, S); label('L',L,SE); label('M',M); label('N',N); [/asy]

Let $M$ be the intersection of $\overbar{AG}$ (Error compiling LaTeX. Unknown error_msg) and $\overbar{BI}$ (Error compiling LaTeX. Unknown error_msg)

and $N$ be the intersection of $\overbar{BI}$ (Error compiling LaTeX. Unknown error_msg) and $\overbar{CJ}$ (Error compiling LaTeX. Unknown error_msg).

More coming^v^

See also

2010 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions