# 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== | ||

+ | We draw a diagram to not lose points: | ||

+ | |||

+ | '''Part a''' | ||

+ | |||

+ | '''Part b''' |

## Revision as of 19:45, 29 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

We draw a diagram to not lose points:

**Part a**

**Part b**