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

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==Problem== | ==Problem== | ||

− | Let <math>\triangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60\ | + | Let <math>\triangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^{\circ}</math>, and let <math>O</math> and <math>H</math> denote the circumcenter and orthocenter of <math>\triangle{ABC}</math>, respectively. |

(a) Prove that line <math>OH</math> intersects both segments <math>AB</math> and <math>AC</math>. | (a) Prove that line <math>OH</math> intersects both segments <math>AB</math> and <math>AC</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>. | (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>. | ||

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

<asy> | <asy> |

## Revision as of 18:23, 30 April 2014

## Problem

Let be a non-equilateral, acute triangle with , 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