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Euler Line Madness
raxu 75
N
2 hours ago
by lakshya2009
Source: TSTST 2015 Problem 2
Let ABC be a scalene triangle. Let
,
and
be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of
intersects
a second time at point
different from A. Define
and
analogously. Prove that the circumcenter of
lies on the Euler line of ABC.
(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)
Proposed by Ivan Borsenco









(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)
Proposed by Ivan Borsenco
75 replies

Own made functional equation
Primeniyazidayi 8
N
2 hours ago
by MathsII-enjoy
Source: own(probably)
Find all functions
such that
for all



8 replies

IMO ShortList 2002, geometry problem 7
orl 110
N
2 hours ago
by SimplisticFormulas
Source: IMO ShortList 2002, geometry problem 7
The incircle
of the acute-angled triangle
is tangent to its side
at a point
. Let
be an altitude of triangle
, and let
be the midpoint of the segment
. If
is the common point of the circle
and the line
(distinct from
), then prove that the incircle
and the circumcircle of triangle
are tangent to each other at the point
.















110 replies
Cute NT Problem
M11100111001Y1R 6
N
2 hours ago
by X.Allaberdiyev
Source: Iran TST 2025 Test 4 Problem 1
A number
is called lucky if it has at least two distinct prime divisors and can be written in the form:
where
are distinct prime numbers that divide
. (Note: it is possible that
has other prime divisors not among
.) Prove that for every prime number
, there exists a lucky number
such that
.

![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)







6 replies
China MO 2021 P6
NTssu 23
N
2 hours ago
by bin_sherlo
Source: CMO 2021 P6
Find
, such that for any
,



23 replies
Prove that the circumcentres of the triangles are collinear
orl 19
N
3 hours ago
by Ilikeminecraft
Source: IMO Shortlist 1997, Q9
Let
be a non-isosceles triangle with incenter
Let
be the smaller circle through
tangent to
and
(the addition of indices being mod 3). Let
be the second point of intersection of
and
Prove that the circumcentres of the triangles
are collinear.











19 replies
An algorithm for discovering prime numbers?
Lukaluce 4
N
3 hours ago
by alexanderhamilton124
Source: 2025 Junior Macedonian Mathematical Olympiad P3
Is there an infinite sequence of prime numbers
such that for every
is satisfied? Explain the answer.


4 replies
Orthocentroidal circle, orthotransversal, concurrent lines
kosmonauten3114 0
3 hours ago
Source: My own
Let
be a scalene oblique triangle, and
a point on the orthocentroidal circle of
(
).
Prove that the orthotransversal of
, trilinear polar of the polar conjugate (
-isoconjugate) of
, Droz-Farny axis of
are concurrent.
The definition of the Droz-Farny axis of
with respect to
is as follows:
For a point
, there exists a pair of orthogonal lines
,
through
such that the midpoints of the 3 segments cut off by
,
from the sidelines of
are collinear. The line through these 3 midpoints is the Droz-Farny axis of
wrt
.




Prove that the orthotransversal of




The definition of the Droz-Farny axis of


For a point









0 replies
