Difference between revisions of "2011 AIME I Problems/Problem 8"

Revision as of 19:36, 3 April 2011

Problem

In triangle $ABC$ , $BC = 23$ , $CA = 27$ , and $AB = 30$ . Points $V$ and $W$ are on $\overline{AC}$ with $V$ on $\overline{AW}$ , points $X$ and $Y$ are on $\overline{BC}$ with $X$ on $\overline{CY}$ , and points $Z$ and $U$ are on $\overline{AB}$ with $Z$ on $\overline{BU}$ . In addition, the points are positioned so that $\overline{UV}\parallel\overline{BC}$ , $\overline{WX}\parallel\overline{AB}$ , and $\overline{YZ}\parallel\overline{CA}$ . Right angle folds are then made along $\overline{UV}$ , $\overline{WX}$ , and $\overline{YZ}$ . The resulting figure is placed on a level floor to make a table with triangular legs. Let $h$ be the maximum possible height of a table constructed from triangle $ABC$ whose top is parallel to the floor. Then $h$ can be written in the form $\frac{k\sqrt{m}}{n}$ , where $k$ and $n$ are relatively prime positive integers and $m$ is a positive integer that is not divisible by the square of any prime. Find $k+m+n$ .

[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(" $A$ ",A[0],N); dot(" $B$ ",B[0],SE); dot(" $C$ ",C[0],SW); dot(" $U$ ",U[0],NE); dot(" $V$ ",V[0],NW); dot(" $W$ ",W[0],NW); dot(" $X$ ",X[0],S); dot(" $Y$ ",Y[0],S); dot(" $Z$ ",Z[0],NE); dot(A[1]); dot(B[1]); dot(C[1]); dot(" $U$ ",U[1],NE); dot(" $V$ ",V[1],NW); dot(" $W$ ",W[1],NW); dot(" $X$ ",X[1],dir(-70)); dot(" $Y$ ",Y[1],dir(250)); dot(" $Z$ ",Z[1],NE);[/asy]

@@ Line 3: / Line 3: @@
-<math>[asy]
+[asy]
 unitsize(1 cm);
 pair translate;
@@ Line 34: / Line 34: @@
 draw (W[1]--C[1]--X[1]);
 draw (Y[1]--B[1]--Z[1]);
-dot("</math>A<math>",A[0],N);
+dot("<math>A</math>",A[0],N);
-dot("</math>B<math>",B[0],SE);
+dot("<math>B</math>",B[0],SE);
-dot("</math>C<math>",C[0],SW);
+dot("<math>C</math>",C[0],SW);
-dot("</math>U<math>",U[0],NE);
+dot("<math>U</math>",U[0],NE);
-dot("</math>V<math>",V[0],NW);
+dot("<math>V</math>",V[0],NW);
-dot("</math>W<math>",W[0],NW);
+dot("<math>W</math>",W[0],NW);
-dot("</math>X<math>",X[0],S);
+dot("<math>X</math>",X[0],S);
-dot("</math>Y<math>",Y[0],S);
+dot("<math>Y</math>",Y[0],S);
-dot("</math>Z<math>",Z[0],NE);
+dot("<math>Z</math>",Z[0],NE);
 dot(A[1]);
 dot(B[1]);
 dot(C[1]);
-dot("</math>U<math>",U[1],NE);
+dot("<math>U</math>",U[1],NE);
-dot("</math>V<math>",V[1],NW);
+dot("<math>V</math>",V[1],NW);
-dot("</math>W<math>",W[1],NW);
+dot("<math>W</math>",W[1],NW);
-dot("</math>X<math>",X[1],dir(-70));
+dot("<math>X</math>",X[1],dir(-70));
-dot("</math>Y<math>",Y[1],dir(250));
+dot("<math>Y</math>",Y[1],dir(250));
-dot("</math>Z<math>",Z[1],NE);[/asy]</math>
+dot("<math>Z</math>",Z[1],NE);[/asy]
 == See also ==
 {{AIME box|year=2011|n=I|num-b=7|num-a=9}}

2011 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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Difference between revisions of "2011 AIME I Problems/Problem 8"

Revision as of 19:36, 3 April 2011

Problem

See also