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

(Created page with "==Problem== Let <math>\triangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60\textdegrees</math>, and let <math>O</math> and <math>H</math> denote the c...")
 
(Solution)
Line 6: Line 6:
 
(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 $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60\textdegrees$ (Error compiling LaTeX. ! Undefined control sequence.), and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$, respectively.

(a) Prove that line $OH$ intersects both segments $AB$ and $AC$.

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

Solution

We draw a diagram to not lose points:

Part a

Part b

Invalid username
Login to AoPS