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

(Solution)
(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'''
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//define dots
 +
A=3*dir(50);
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B=(0,0);
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C=right*2.76481496;
  
'''Part b'''
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O=circumcenter(A,B,C);
 +
H=orthocenter(A,B,C);
 +
 
 +
i1=2*O-H;
 +
i2=2*i1-O;
 +
i3=2*H-O;
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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);
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dot(A);
 +
dot(B);
 +
dot(C);
 +
dot(O);
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dot(H);
 +
draw(A--B--C--cycle);
 +
 
 +
//label
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label("$A$",A,N);
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label("$B$",B,SW);
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label("$C$",C,SE);
 +
label("$P$",P,NW);
 +
label("$Q$",Q,NE);
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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 $\triangle{ABC}$ 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

[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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