by MathMaxGreat, Jul 11, 2025, 2:37 AM 
, we can post new and original problems, my idea is to make an competition like 
, 6 problems for 2 rounds by RANDOM__USER, Jul 9, 2025, 2:36 PM 
be an arbitrary point on the side 
of triangle 
. Let 
and 
be the intersections of the lines through , parallel to 
and 
, with and , respectively. Let 
be the intersection point of the circumcircle of 
with the circumcircle of . Let 
be the midpoint of , and let 
be the intersection point of line 
with the circumcircle of . Prove that , , and 
are collinear.![[asy]
pair A = (3.66190,4.28553);
pair C = (0.,0.);
pair B = (5.,0.);
pair D = (3.10844,0.);
pair F = (2.27656,2.66426);
pair E = (4.49378,1.62126);
pair G = (5.15982,2.85311);
pair X = (2.13167,-1.35853);
pair M = (2.5,0.);
import graph;
size(12cm);
pen zzttqq = rgb(0.6,0.2,0.);
draw(A--B--C--cycle, linewidth(0.6) + zzttqq);
draw(circle((2.5,1.57107), 2.95267), linewidth(0.6));
draw(A--B, linewidth(0.6) + zzttqq);
draw(B--C, linewidth(0.6) + zzttqq);
draw(C--A, linewidth(0.6) + zzttqq);
draw(D--E, linewidth(0.6));
draw(D--F, linewidth(0.6));
draw(circle((3.71290,2.83944), 1.44698), linewidth(0.6));
draw(G--X, linewidth(0.6)+deepblue);
draw(X--A, linewidth(0.6));
dot("$A$", A, dir(68));
dot("$C$", C, dir(207));
dot("$B$", B, dir(298));
dot("$D$", D, dir(273));
dot("$F$", F, dir(188));
dot("$E$", E, dir(-20));
dot("$G$", G, dir(1));
dot("$X$", X, dir(248));
dot("$M$", M, dir(120));
[/asy]](http://latex.artofproblemsolving.com/d/f/b/dfb73458178cbcef1c1beb77b30c1a84eaa123d0.png)
by fattypiggy123, Mar 11, 2019, 8:45 AM 
good if for any distinct positive integers 
, one has
Call a positive integer 
to be 
-good if there exists a good sequence such that 
. Does there exists a such that there are exactly 
-good positive integers?by 62861, Dec 11, 2017, 5:00 PM ![$f\colon \mathbb{Z}^2 \to [0, 1]$](http://latex.artofproblemsolving.com/3/2/e/32e14eccb91bbe356013e2a6e5a37a2b54223e08.png)
such that for any integers 
and 
,![\[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\]](http://latex.artofproblemsolving.com/9/a/0/9a0933a0d4c566c8201d1a26053bc992a21d4d45.png)
by Amir Hossein, Oct 4, 2011, 3:18 AM 
. Show that the sum of the lengths of each pair of opposite sides of is equal.by cmtappu96, Dec 5, 2010, 11:38 AM 
be a triangle in which 
. Let 
and 
be the bisectors of 
and 
with 
on 
and 
on 
. Let 
be the reflection of 
in line 
. Prove that lies on 
.by Amir Hossein, Sep 1, 2010, 8:12 AM 
be the incenter, centroid, and circumcenter of the nonisosceles triangle . Prove that 
if and only if 
.by Goutham, May 7, 2010, 6:50 PM be an acute angled triangle satisfying the conditions 
and 
. Denote by 
and 
the circumcentre and orthocentre, respectively, of the triangle 
Suppose that the circumcircle of the triangle 
intersects the line at different from , and the circumcircle of the triangle 
intersects the line at 
different from 
Prove that the circumcentre of the triangle 
lies on the line 
.by Valentin Vornicu, Jul 13, 2006, 11:44 AM 
of integers such that ![\[1+2^{x}+2^{2x+1}= y^{2}.\]](http://latex.artofproblemsolving.com/9/c/8/9c81ffa59a374578a1dba99ad772902ea17ebadb.png)
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