Mock AIME 1 2007-2008 Problems/Problem 11
Problem
is inscribed inside 
such that 
lie on 
, respectively. The circumcircles of 
have centers 
, respectively. Also, 
, and 
. The length of 
can be written in the form 
, where 
and 
are relatively prime integers. Find 
.
Solution
size(150); defaultpen(linewidth(0.8)); import markers; import geometry_dev;
pair B = (0,0), C = (25,0), A = (578/50,19.8838);
draw(A--B--C--cycle);
label("
",B,SW); label("
",C,SE); label("
",A,N);
pair D = (13,0), E = (11*A + 13*C)/24, F = (12*B + 11*A)/23;
draw(D--E--F--cycle);
label("
",D,dir(-90));
label("
",E,dir(0));
label("
",F,dir(180));
draw(A--E,StickIntervalMarker(1,3,size=6));draw(B--D,StickIntervalMarker(1,3,size=6)); draw(F--B,StickIntervalMarker(1,2,size=6)); draw(E--C,StickIntervalMarker(1,2,size=6)); draw(A--F,StickIntervalMarker(1,1,size=6)); draw(C--D,StickIntervalMarker(1,1,size=6));
label("24",A--C,5*dir(0)); label("25",B--C,5*dir(-90)); label("23",B--A,5*dir(180));
</asy>From adjacent sides, the following relationships can be derived:
$\begin{align*} DC &= EC + 1\\ AE &= AF + 1\\ BD &= BF + 2 \end{align*}$ (Error compiling LaTeX. Unknown error_msg)
Since 
, and 
, 
. Thus, 
. 
. Thus, 
. Thus, the answer is 
.
See also
| Mock AIME 1 2007-2008 (Problems, Source) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
