![[asy]
size(10cm);
import olympiad;
import geometry;
pair A = dir(119);
pair A_prime = -A;
pair E = reflect((0,0), dir(90)) * A_prime;
pair S = dir(90);
pair D = 0.6*E + 0.4*A;
pair B = 2 * foot((0,0), D, S) - S;
pair P = 2 * foot((0,0), A_prime, D) - A_prime;
pair Q = reflect((0,0), (P+A_prime)/2) * S;
pair M = -S;
pair X = extension(E,Q,S,D);
pair Y = extension(P,Q,A,M);
pair L = extension(D+E-A_prime, D, B,E);
fill(X--M--Q--cycle, paleblue);
fill(A--P--S--cycle, paleblue);
draw(unitcircle, linewidth(1));
draw(S--X, red+linewidth(1.2));
draw(P--A_prime, linewidth(0.8));
draw(circle(B,D,P), linewidth(0.8));
draw(P--Q, linewidth(0.8));
draw(X--M--Q--cycle, blue+linewidth(0.8));
draw(A--P--S--cycle, blue+linewidth(0.8));
draw(A--M, linewidth(0.8));
draw(A--E--A_prime, gray+linewidth(0.7));
draw(S--M, gray+linewidth(0.7));
draw(D--L, gray+linewidth(0.7));
dot("$A$", A, dir(118));
dot("$A'$", A_prime, dir(-62));
dot("$E$", E, dir(-84));
dot("$S$", S, dir(89));
dot("$D$", D, dir(16));
dot("$B$", B, dir(-4));
dot("$P$", P, dir(133));
dot("$Q$", Q, dir(17));
dot("$M$", M, dir(-82));
dot("$X$", X, dir(-139));
dot("$Y$", Y, dir(143));
dot("$L$", L, dir(165));
[/asy]](https://latex.artofproblemsolving.com/b/5/b/b5b3a7a42bfd5da332a8282ef2402a6997a5a690.png)
Let











- Since
, we get that
are collinear.
- Pascal on
gives
,
, and
are collinear.
- We have
so
. Thus,
and
are homothetic at center
, so
are collinear.
Hence, are collinear.
clearly lies on this line, done.
Solution 2
Problem 2.
Let ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() proposed by Tiago MourĂ£o and Nuno Arala, Colombia Solution 1
![]() Let ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
Hence, Solution 2 Let
![]() ![]() ![]() ![]() be ![]() ![]() So ![]() ![]() We it suffices to show that ![]() It is sufficient that is ![]() But it is true that ![]() So finally it is sufficient ![]() ![]() ![]() Therefore from ![]() ![]() ![]() which is direct. The proof is complete.
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