# Difference between revisions of "2014 USAJMO Problems/Problem 2"

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(b) Line <math>OH</math> intersects segments <math>AB</math> and <math>AC</math> at <math>P</math> and <math>Q</math>, respectively. Denote by <math>s</math> and <math>t</math> the respective areas of triangle <math>APQ</math> and quadrilateral <math>BPQC</math>. Determine the range of possible values for <math>s/t</math>. | (b) Line <math>OH</math> intersects segments <math>AB</math> and <math>AC</math> at <math>P</math> and <math>Q</math>, respectively. Denote by <math>s</math> and <math>t</math> the respective areas of triangle <math>APQ</math> and quadrilateral <math>BPQC</math>. Determine the range of possible values for <math>s/t</math>. | ||

==Solution== | ==Solution== | ||

− | + | <asy> | |

+ | import olympiad; | ||

+ | unitsize(1inch); | ||

+ | pair A,B,C,O,H,P,Q,i1,i2,i3,i4; | ||

− | + | //define dots | |

+ | A=3*dir(50); | ||

+ | B=(0,0); | ||

+ | C=right*2.76481496; | ||

− | + | O=circumcenter(A,B,C); | |

+ | H=orthocenter(A,B,C); | ||

+ | |||

+ | i1=2*O-H; | ||

+ | i2=2*i1-O; | ||

+ | i3=2*H-O; | ||

+ | i4=2*i3-H; | ||

+ | //These points are for extending PQ. DO NOT DELETE! | ||

+ | |||

+ | P=intersectionpoint(i2--i4,A--B); | ||

+ | Q=intersectionpoint(i2--i4,A--C); | ||

+ | |||

+ | //draw | ||

+ | dot(P); | ||

+ | dot(Q); | ||

+ | draw(P--Q); | ||

+ | dot(A); | ||

+ | dot(B); | ||

+ | dot(C); | ||

+ | dot(O); | ||

+ | dot(H); | ||

+ | draw(A--B--C--cycle); | ||

+ | |||

+ | //label | ||

+ | label("$A$",A,N); | ||

+ | label("$B$",B,SW); | ||

+ | label("$C$",C,SE); | ||

+ | label("$P$",P,NW); | ||

+ | label("$Q$",Q,NE); | ||

+ | label("$O$",O,N); | ||

+ | label("$H$",H,N); | ||

+ | //change O and H label positions if interfering with other lines to be added | ||

+ | |||

+ | //further editing: ABCPQOH are the dots to be further used. i1,i2,i3,i4 are for drawing assistence and are not to be used | ||

+ | </asy> |

## Revision as of 18:21, 30 April 2014

## Problem

Let be a non-equilateral, acute triangle with $\angle A=60\textdegrees$ (Error compiling LaTeX. ! Undefined control sequence.), and let and denote the circumcenter and orthocenter of , respectively.

(a) Prove that line intersects both segments and .

(b) Line intersects segments and at and , respectively. Denote by and the respective areas of triangle and quadrilateral . Determine the range of possible values for .

## Solution