by Nuran2010, Apr 29, 2025, 4:52 PM 
is a positive integer. Call all positive divisors of which are different from 
and beautiful divisors.We call a special number when it has at least 
beautiful divisors and difference of any beautiful divisors divides as well. Find all special numbers.by MathSaiyan, Mar 17, 2025, 2:00 PM 
and 
be positive integers. For a connected simple graph 
on vertices and edges, we consider the number 
of orientations of (all of) its edges so that, in the resulting directed graph, every vertex has even outdegree. only depends on and , and determine its value.by Supercali, Dec 20, 2022, 12:08 PM 
satisfying
for all integers 
.by darij grinberg, Mar 11, 2022, 1:01 PM 
be a triangle with orthocenter 
and circumcenter 
. Let 
be a point in the plane such that 
. Let 
and 
be the reflections of in the lines 
and 
, respectively. Let 
be the orthogonal projection of onto . Let 
be the orthogonal projection of onto . Assume that 
and 
. Prove that 
.![[asy]
import olympiad;
unitsize(30);
pair A,B,C,H,O,P,Q,R,Y,Z,Q2,R2,P2;
A = (-14.8, -6.6);
B = (-10.9, 0.3);
C = (-3.1, -7.1);
O = circumcenter(A,B,C);
H = orthocenter(A,B,C);
P = 1.2 * H - 0.2 * A;
Q = reflect(A, C) * P;
R = reflect(A, B) * P;
Y = foot(R, C, A);
Z = foot(Q, A, B);
P2 = foot(A, B, C);
Q2 = foot(P, C, A);
R2 = foot(P, A, B);
draw(B--(1.6*A-0.6*B));
draw(B--C--A);
draw(P--R, blue);
draw(R--Y, red);
draw(P--Q, blue);
draw(Q--Z, red);
draw(A--P2, blue);
draw(O--H, darkgreen+linewidth(1.2));
draw((1.4*Z-0.4*Y)--(4.6*Y-3.6*Z), red+linewidth(1.2));
draw(rightanglemark(R,Y,A,10), red);
draw(rightanglemark(Q,Z,B,10), red);
draw(rightanglemark(C,Q2,P,10), blue);
draw(rightanglemark(A,R2,P,10), blue);
draw(rightanglemark(B,P2,H,10), blue);
label("$\textcolor{blue}{H}$",H,NW);
label("$\textcolor{blue}{P}$",P,N);
label("$A$",A,W);
label("$B$",B,N);
label("$C$",C,S);
label("$O$",O,S);
label("$\textcolor{blue}{Q}$",Q,E);
label("$\textcolor{blue}{R}$",R,W);
label("$\textcolor{red}{Y}$",Y,S);
label("$\textcolor{red}{Z}$",Z,NW);
dot(A, filltype=FillDraw(black));
dot(B, filltype=FillDraw(black));
dot(C, filltype=FillDraw(black));
dot(H, filltype=FillDraw(blue));
dot(P, filltype=FillDraw(blue));
dot(Q, filltype=FillDraw(blue));
dot(R, filltype=FillDraw(blue));
dot(Y, filltype=FillDraw(red));
dot(Z, filltype=FillDraw(red));
dot(O, filltype=FillDraw(black));
[/asy]](http://latex.artofproblemsolving.com/0/0/c/00c2d50d0fec3d1019ecee07ae97d97f179c9dd5.png)
by mathlogician, Sep 11, 2020, 11:52 PM 
be a function that satisfies
for all positive integer pairs 
Find all possible functions 
as the set of all positive integers.)by dchenmathcounts, May 23, 2020, 11:00 PM 
be a cyclic quadrilateral. A circle centered at passes through 
and 
and meets lines 
and 
again at points 
and 
(distinct from 
). Let denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle and 
are similar.by parmenides51, Mar 26, 2020, 5:08 PM so that 
is an integer.by Djile, Apr 8, 2013, 3:13 PM 
, and be midpoints of sides 
and , respectively, and let be circumcenter of acute-angled triangle . Circumcircles of triangles 
and 
intersect at two different points 
and inside of triangle . Prove that ![\[\angle BAX=\angle CAY.\]](http://latex.artofproblemsolving.com/3/9/9/39910fdd000cbed820b33ac08a8200ca06ed8193.png)
September 2012
May 2012
March 2012
February 2012
January 2012
December 2011
November 2011
October 2011
September 2011
August 2011
July 2011
May 2011
April 2011
March 2011
February 2011
December 2010
November 2010
October 2010
September 2010
August 2010
July 2010
June 2010
May 2010
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
and 
.
