hard problem
by Cobedangiu, Apr 21, 2025, 1:51 PM
Counting graph theory
by MathSaiyan, Mar 17, 2025, 2:00 PM
Let
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.
Show that
only depends on
and
, and determine its value.






Show that



2 variable functional equation in integers
by Supercali, Dec 20, 2022, 12:08 PM
Find all functions
satisfying
for all integers
.



Another perpendicular to the Euler line
by darij grinberg, Mar 11, 2022, 1:01 PM
Let
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]](//latex.artofproblemsolving.com/0/0/c/00c2d50d0fec3d1019ecee07ae97d97f179c9dd5.png)



















![[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)
Integer Functional Equation
by mathlogician, Sep 11, 2020, 11:52 PM
Let
be a function that satisfies
for all positive integer pairs
Find all possible functions 
(Here, we define
as the set of all positive integers.)




(Here, we define

H not needed
by dchenmathcounts, May 23, 2020, 11:00 PM
Let
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.
Robin Son
















Robin Son
This post has been edited 2 times. Last edited by v_Enhance, Oct 25, 2020, 6:01 AM
Reason: backdate
Reason: backdate
\sqrt{(1^2+2^2+...+n^2)/n}$ is an integer.
by parmenides51, Mar 26, 2020, 5:08 PM
Find the smallest positive integer
so that
is an integer.


This post has been edited 1 time. Last edited by parmenides51, Mar 28, 2020, 3:42 AM
Intersection of circumcircles of MNP and BOC
by Djile, Apr 8, 2013, 3:13 PM
Let
,
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 ![\[\angle BAX=\angle CAY.\]](//latex.artofproblemsolving.com/3/9/9/39910fdd000cbed820b33ac08a8200ca06ed8193.png)












![\[\angle BAX=\angle CAY.\]](http://latex.artofproblemsolving.com/3/9/9/39910fdd000cbed820b33ac08a8200ca06ed8193.png)
Archives
















Shouts
Submit
98 shouts
Contributors
Tags
About Owner
- Posts: 2106
- Joined: Aug 20, 2016
Blog Stats
- Blog created: Mar 28, 2020
- Total entries: 61
- Total visits: 4962
- Total comments: 146
Search Blog