# Difference between revisions of "2013 USAJMO Problems/Problem 5"

(→Problem) |
(→Solution) |
||

Line 3: | Line 3: | ||

Quadrilateral <math>XABY</math> is inscribed in the semicircle <math>\omega</math> with diameter <math>XY</math>. Segments <math>AY</math> and <math>BX</math> meet at <math>P</math>. Point <math>Z</math> is the foot of the perpendicular from <math>P</math> to line <math>XY</math>. Point <math>C</math> lies on <math>\omega</math> such that line <math>XC</math> is perpendicular to line <math>AZ</math>. Let <math>Q</math> be the intersection of segments <math>AY</math> and <math>XC</math>. Prove that <cmath>\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.</cmath> | Quadrilateral <math>XABY</math> is inscribed in the semicircle <math>\omega</math> with diameter <math>XY</math>. Segments <math>AY</math> and <math>BX</math> meet at <math>P</math>. Point <math>Z</math> is the foot of the perpendicular from <math>P</math> to line <math>XY</math>. Point <math>C</math> lies on <math>\omega</math> such that line <math>XC</math> is perpendicular to line <math>AZ</math>. Let <math>Q</math> be the intersection of segments <math>AY</math> and <math>XC</math>. Prove that <cmath>\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.</cmath> | ||

− | ==Solution== | + | ==Problem== |

+ | Please copy and paste the problem here! | ||

+ | |||

+ | ==Solution 1== | ||

+ | Let us use coordinates. Let O, the center of the circle, be (0,0). WLOG the radius of the circle is 1, so set Y (1,0) and X (-1,0). Also, for arbitrary constants <math>a</math> and <math>b</math> set A <math>(\cos a, \sin a)</math> and B <math>(\cos b, \sin b)</math>. Now, let's use our coordinate tools. It is easily derived that the equation of <math>BX</math> is <math>y = \frac{\sin b}{1 + \cos b}(x + 1) = v(x+1)</math> and the equation of <math>AY</math> is <math>y = \frac{\sin a}{1 - cos a}(x - 1) = u(x-1)</math>, where <math>u</math> and <math>v</math> are defined appropriately. Thus, by equating the y's in the equation we find the intersection of these lines, <math>P</math>, is <math>(\frac{u-v}{u+v}, \frac{2uv}{u+v})</math>. Also, <math>Z(\frac{u-v}{u+v}, 0)</math>. It shall be left to the reader to find the slope of <math>AZ</math>, the coordinates of Q and C, and use the distance formula to verify that <math>\frac{BY}{XP} + \frac{CY}{XQ} = \frac{AY}{AX}</math>. | ||

+ | |||

+ | ==Solution 2== | ||

+ | There should be a geometric solution.... | ||

+ | |||

+ | {{Solution}} | ||

+ | |||

{{MAA Notice}} | {{MAA Notice}} |

## Revision as of 22:29, 28 April 2014

## Contents

## Problem

Quadrilateral is inscribed in the semicircle with diameter . Segments and meet at . Point is the foot of the perpendicular from to line . Point lies on such that line is perpendicular to line . Let be the intersection of segments and . Prove that

## Problem

Please copy and paste the problem here!

## Solution 1

Let us use coordinates. Let O, the center of the circle, be (0,0). WLOG the radius of the circle is 1, so set Y (1,0) and X (-1,0). Also, for arbitrary constants and set A and B . Now, let's use our coordinate tools. It is easily derived that the equation of is and the equation of is , where and are defined appropriately. Thus, by equating the y's in the equation we find the intersection of these lines, , is . Also, . It shall be left to the reader to find the slope of , the coordinates of Q and C, and use the distance formula to verify that .

## Solution 2

There should be a geometric solution....

*This problem needs a solution. If you have a solution for it, please help us out by adding it.*

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.