2011 AIME I Problems/Problem 8
Problem
In triangle 
, 
, 
, and 
. Points 
and 
are on 
with on 
, points 
and 
are on 
with on 
, and points 
and 
are on 
with on 
. In addition, the points are positioned so that 
, 
, and 
. Right angle folds are then made along 
, 
, and 
. The resulting figure is placed on a level floor to make a table with triangular legs. Let 
be the maximum possible height of a table constructed from triangle whose top is parallel to the floor. Then can be written in the form 
, where 
and 
are relatively prime positive integers and 
is a positive integer that is not divisible by the square of any prime. Find 
.
[asy]
unitsize(1 cm);
pair translate;
pair[] A, B, C, U, V, W, X, Y, Z;
A[0] = (1.5,2.8);
B[0] = (3.2,0);
C[0] = (0,0);
U[0] = (0.69*A[0] + 0.31*B[0]);
V[0] = (0.69*A[0] + 0.31*C[0]);
W[0] = (0.69*C[0] + 0.31*A[0]);
X[0] = (0.69*C[0] + 0.31*B[0]);
Y[0] = (0.69*B[0] + 0.31*C[0]);
Z[0] = (0.69*B[0] + 0.31*A[0]);
translate = (7,0);
A[1] = (1.3,1.1) + translate;
B[1] = (2.4,-0.7) + translate;
C[1] = (0.6,-0.7) + translate;
U[1] = U[0] + translate;
V[1] = V[0] + translate;
W[1] = W[0] + translate;
X[1] = X[0] + translate;
Y[1] = Y[0] + translate;
Z[1] = Z[0] + translate;
draw (A[0]--B[0]--C[0]--cycle);
draw (U[0]--V[0],dashed);
draw (W[0]--X[0],dashed);
draw (Y[0]--Z[0],dashed);
draw (U[1]--V[1]--W[1]--X[1]--Y[1]--Z[1]--cycle);
draw (U[1]--A[1]--V[1],dashed);
draw (W[1]--C[1]--X[1]);
draw (Y[1]--B[1]--Z[1]);
dot("
",A[0],N);
dot("
",B[0],SE);
dot("
",C[0],SW);
dot("",U[0],NE);
dot("",V[0],NW);
dot("",W[0],NW);
dot("",X[0],S);
dot("",Y[0],S);
dot("",Z[0],NE);
dot(A[1]);
dot(B[1]);
dot(C[1]);
dot("",U[1],NE);
dot("",V[1],NW);
dot("",W[1],NW);
dot("",X[1],dir(-70));
dot("",Y[1],dir(250));
dot("",Z[1],NE);[/asy]
See also
| 2011 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
