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Difference between revisions of "2006 iTest Problems/Problem U8"

(Created page with "==Problem== Let <math>T = TNFTPP</math>, and let <math>S</math> be the sum of the digits of <math>T</math>. Cyclic quadrilateral <math>ABCD</math> has side lengths <mat...")
 
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==Problem==
 
==Problem==
  
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Let <math>T  =  TNFTPP</math>, and let  <math>S</math> be the sum of the digits of  <math>T</math>. Cyclic quadrilateral  <math>ABCD</math> has side lengths  <math>AB  =  5</math>,  <math>BC  =  2</math>,  <math>CD  =  3</math>, and  <math>DA  =  10</math>. Let  <math>M</math> and  <math>N</math> be the midpoints of sides  <math>AD</math> and <math>BC</math>. The diagonals  <math>AC</math> and  <math>BD</math> intersect  <math>MN</math> at  <math>P</math> and  <math>Q</math> respectively.  <math>\frac{PQ}{MN}</math> can be expressed as <math>\frac{m}{n}</math> where  <math>m</math> and  <math>n</math> are relatively prime positive integers. Determine  <math>m  +  n</math>.
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Cyclic quadrilateral  <math>ABCD</math> has side lengths  <math>AB  =  9</math>,  <math>BC  =  2</math>,  <math>CD  =  3</math>, and  <math>DA  =  10</math>. Let  <math>M</math> and  <math>N</math> be the midpoints of sides  <math>AD</math> and <math>BC</math>. The diagonals  <math>AC</math> and  <math>BD</math> intersect  <math>MN</math> at  <math>P</math> and  <math>Q</math> respectively.  <math>\frac{PQ}{MN}</math> can be expressed as <math>\frac{m}{n}</math> where  <math>m</math> and  <math>n</math> are relatively prime positive integers. Determine  <math>m  +  n</math>.
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[[Category: Intermediate Geometry Problems]]

Latest revision as of 17:57, 17 December 2021

Problem

Cyclic quadrilateral $ABCD$ has side lengths $AB = 9$, $BC = 2$, $CD = 3$, and $DA = 10$. Let $M$ and $N$ be the midpoints of sides $AD$ and $BC$. The diagonals $AC$ and $BD$ intersect $MN$ at $P$ and $Q$ respectively. $\frac{PQ}{MN}$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine $m + n$.

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