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

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(→‎Diagram)
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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:
+
<asy>
 +
import olympiad;
 +
unitsize(1inch);
 +
pair A,B,C,O,H,P,Q,i1,i2,i3,i4;
  
−
'''Part a'''
+
//define dots
 +
A=3*dir(50);
 +
B=(0,0);
 +
C=right*2.76481496;
  
−
'''Part b'''
+
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 19:21, 30 April 2014

Problem

Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60\textdegrees$ (Error compiling LaTeX. Unknown error_msg), 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

[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]

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