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Difference between revisions of "2013 Mock AIME I Problems/Problem 5"

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== Solution ==
 
== Solution ==
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<asy>
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import geometry;
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// Defining Points
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point O = origin;
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point B = (1,0);
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point A = dir(115.583);
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point C = dir(-115.583);
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point D = dir(-165.638);
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point M;
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// Circle
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draw(circle(O, 1));
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// Quadrilateral and Diagonals
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draw(A--B--C--D--cycle);
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draw(A--C);
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draw(B--D);
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// Defining M
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pair[] m = intersectionpoints((A--C),(B--D));
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M = m[0];
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// Labelling Points
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dot(A);
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label("A",A,NW);
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dot(B);
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label("B",B,E);
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dot(C);
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label("C",C,SW);
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dot(D);
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label("D",D,WSW);
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dot(M);
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label("M",M,NE);
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// Length Labels
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label("$3$", midpoint(D--M), NNW);
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label("$8$", midpoint(M--B), NNW);
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label("$6$", midpoint(A--M), E);
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label("$4$", midpoint(C--M), E);
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label("$2x$", midpoint(B--C), SE);
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label("$x$", midpoint(C--D), NE);
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</asy>
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<math>\boxed{037}</math>.
 
<math>\boxed{037}</math>.
  

Revision as of 11:11, 30 July 2024

Problem

In quadrilateral $ABCD$, $AC\cap BD=M$. Also, $MA=6, MB=8, MC=4, MD=3$, and $BC=2CD$. The perimeter of $ABCD$ can be expressed in the form $\frac{p\sqrt{q}}{r}$ where $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime number. Find $p+q+r$.

Solution

[asy] import geometry; // Defining Points point O = origin; point B = (1,0); point A = dir(115.583); point C = dir(-115.583); point D = dir(-165.638); point M; // Circle draw(circle(O, 1)); // Quadrilateral and Diagonals draw(A--B--C--D--cycle); draw(A--C); draw(B--D); // Defining M pair[] m = intersectionpoints((A--C),(B--D)); M = m[0]; // Labelling Points dot(A); label("A",A,NW); dot(B); label("B",B,E); dot(C); label("C",C,SW); dot(D); label("D",D,WSW); dot(M); label("M",M,NE); // Length Labels label("$3$", midpoint(D--M), NNW); label("$8$", midpoint(M--B), NNW); label("$6$", midpoint(A--M), E); label("$4$", midpoint(C--M), E); label("$2x$", midpoint(B--C), SE); label("$x$", midpoint(C--D), NE); [/asy]

$\boxed{037}$.

See also

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