2002 IMO Problems/Problem 2

Problem

$\text{BC is a diameter of a circle center O. A is any point on the circle with } \angle AOC \not\le 60^\circ$

$\text{EF is the chord which is the perpendicular bisector of AO. D is the midpoint of the minor arc AB. The line through}$

$\text{O parallel to AD meets AC at J. Show that J is the incenter of triangle CEF.}$

Solution

$\text{By construction, AEOF is a rhombus with } 60^\circ - 120^\circ \text{angles}$

$\text{ Consequently, we may set } s = AO = AE = AF = EO = EF$

$\documentclass{article}

\usepackage[pdftex]{graphicx} \usepackage{asymptote} \begin{document} Hello. I like to make pics with Asymptote like this one: \begin{figure}[h]

 \begin{asy}
   import graph;
   size(1inch);
   filldraw(circle((0,0),1),yellow,black);
   fill(circle((-.3,.4),.1),black);
   fill(circle((.3,.4),.1),black);
   draw(arc((0,0),.5,-140,-40));
 \end{asy}

\end{figure} \par It makes me happy, since I can still type my normal LaTeX stuff around it: $\int_0^{\pi}{\sin{x}}\,dx=2$ \end{document}\documentclass{article} \usepackage[pdftex]{graphicx} \usepackage{asymptote} \begin{document} Hello. I like to make pics with Asymptote like this one: \begin{figure}[h]

 \begin{asy}
   import graph;
   size(1inch);
   filldraw(circle((0,0),1),yellow,black);
   fill(circle((-.3,.4),.1),black);
   fill(circle((.3,.4),.1),black);
   draw(arc((0,0),.5,-140,-40));
 \end{asy}

\end{figure} \par It makes me happy, since I can still type my normal LaTeX stuff around it: $\int_0^{\pi}{\sin{x}}\,dx=2$ \end{document}$ (Error compiling LaTeX. Unknown error_msg)

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2002 IMO Problems/Problem 2

Problem

Solution