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Collinearity of intersection points in a triangle
MathMystic33 2
N
by AylyGayypow009
Source: 2025 Macedonian Team Selection Test P1
On the sides of the triangle 
lie the following points: 
and 
on 
, 
on 
, and 
on 
. Let
![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](//latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet at 
. Prove that the points 
, , and 
are collinear.
lie the following points: 
and 
on 
, 
on 
, and 
on 
. Let![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](http://latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet at 
. Prove that the points 
, , and 
are collinear.
2 replies
Parallel lines in incircle configuration
GeorgeRP 1
N
by Double07
Source: Bulgaria IMO TST 2025 P1
Let 
be the incenter of triangle 
. Let 
, 
, and 
be the orthocenters of triangles 
, 
, and 
, respectively. Prove that the lines through , , and , parallel to 
, 
, and 
, respectively, are concurrent.
be the incenter of triangle 
. Let 
, 
, and 
be the orthocenters of triangles 
, 
, and 
, respectively. Prove that the lines through , , and , parallel to 
, 
, and 
, respectively, are concurrent.
1 reply
1 viewing
Proving that these are concyclic.
Acrylic3491 0
In 
, points 
and 
are isogonal conjugates. The tangent to 
at and the tangent to 
at Q, meet at 
. 
intersects 
at 
. Prove that points ,, and are concyclic.
Any hints on this ?
, points 
and 
are isogonal conjugates. The tangent to 
at and the tangent to 
at Q, meet at 
. 
intersects 
at 
. Prove that points ,, and are concyclic.Any hints on this ?
0 replies
. Prove that
W here 
BMO Shortlist 2021 G2
Lukaluce 6
N
by tilya_TASh
Source: BMO Shortlist 2021
Let and 
be the incenter and the circumcenter of a triangle 
, respectively, and let
be the exterior bisector of angle 
. The line through perpendicular to 
meets the
lines
and at points and , respectively. Prove that 
.
and 
be the incenter and the circumcenter of a triangle 
, respectively, and let
be the exterior bisector of angle 
. The line through perpendicular to 
meets thelines
and at points and , respectively. Prove that 
.6 replies
and 
Show that
ISI UGB 2025 P2
SomeonecoolLovesMaths 9
N
by SatisfiedMagma
Source: ISI UGB 2025 P2
If the interior angles of a triangle satisfy the equality, 
prove that the triangle must have a right angle.
satisfy the equality, 
prove that the triangle must have a right angle.
9 replies
.
.![\[\sum ab +\sum ab^2 \le 6\]](http://latex.artofproblemsolving.com/9/f/6/9f632ab1e6df5029bcd4416eb06623123ad2cd8c.png)
Number theory
EeEeRUT 1
N
by lgx57
Source: Thailand MO 2025 P10
Let 
be a positive integer. Show that there exist a polynomial 
with integer coefficient that satisfy the following
[list]
[*]Degree of is at most 
[*]
for each 
[/list]
be a positive integer. Show that there exist a polynomial 
with integer coefficient that satisfy the following[list]
[*]Degree of
is at most 
[*]
for each 
[/list]
1 reply
