such that 
Y by Adventure10, Mango247
Prove that orthotransversal of incenter wrt reference triangle and wrt incentral triangle are symmetric through such this incenter.
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Nordic 2025 P3
anirbanbz 8
N
by Primeniyazidayi
Source: Nordic 2025
Let 
be an acute triangle with orthocenter 
and circumcenter 
. Let 
and 
be points on the line segments 
and 
respectively such that 
is a parallelogram. Prove that 
.
be an acute triangle with orthocenter 
and circumcenter 
. Let 
and 
be points on the line segments 
and 
respectively such that 
is a parallelogram. Prove that 
.
8 replies
A number theory about divisors which no one fully solved at the contest
nAalniaOMliO 21
N
by nAalniaOMliO
Source: Belarusian national olympiad 2024
Let's call a pair of positive integers 
interesting if 
is composite and for every divisor 
of at least one of 
and 
is also a divisor of
Find the number of interesting pairs with 
M. Karpuk
interesting if 
is composite and for every divisor 
of at least one of 
and 
is also a divisor of Find the number of interesting pairs
with 
M. Karpuk
21 replies
2025 Caucasus MO Seniors P2
BR1F1SZ 3
N
by X.Luser
Source: Caucasus MO
Let be a triangle, and let 
and 
be points on segment symmetric with respect to the midpoint of . Let 
denote the circle passing through and tangent to line at 
. Similarly, let 
denote the circle passing through and tangent to line 
at 
. Let the circles and intersect again at point 
(
). Prove that 
.
be a triangle, and let 
and 
be points on segment symmetric with respect to the midpoint of . Let 
denote the circle passing through and tangent to line at 
. Similarly, let 
denote the circle passing through and tangent to line 
at 
. Let the circles and intersect again at point 
(
). Prove that 
.
3 replies
IMO Shortlist 2010 - Problem G1
Amir Hossein 130
N
by LeYohan
Let be an acute triangle with 
the feet of the altitudes lying on 
respectively. One of the intersection points of the line 
and the circumcircle is 
The lines 
and 
meet at point 
Prove that 
Proposed by Christopher Bradley, United Kingdom
be an acute triangle with 
the feet of the altitudes lying on 
respectively. One of the intersection points of the line 
and the circumcircle is 
The lines 
and 
meet at point 
Prove that 
Proposed by Christopher Bradley, United Kingdom
130 replies
CGMO6: Airline companies and cities
v_Enhance 13
N
by Marcus_Zhang
Source: 2012 China Girl's Mathematical Olympiad
There are cities, 
airline companies in a country. Between any two cities, there is exactly one -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of .
cities, 
airline companies in a country. Between any two cities, there is exactly one -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of .
13 replies
nice problem
hanzo.ei 0
Source: I forgot
Let triangle be inscribed in the circumcircle 
and circumscribed about the incircle 
, with 
. The incircle touches the sides , 
, and at 
, , and , respectively. A line through 
, perpendicular to 
, intersects , , and at 
, 
, and 
, respectively. The line meets at 
(distinct from ). The circumcircle of triangle 
intersects at 
(distinct from ). Let 
be the midpoint of the arc 
of . The line cuts segments and 
at 
and 
, respectively, and the tangents to the circle 
at and intersect at 
. Prove that 
.
be inscribed in the circumcircle 
and circumscribed about the incircle 
, with 
. The incircle touches the sides , 
, and at 
, , and , respectively. A line through 
, perpendicular to 
, intersects , , and at 
, 
, and 
, respectively. The line meets at 
(distinct from ). The circumcircle of triangle 
intersects at 
(distinct from ). Let 
be the midpoint of the arc 
of . The line cuts segments and 
at 
and 
, respectively, and the tangents to the circle 
at and intersect at 
. Prove that 
.
0 replies
Find a given number of divisors of ab
proglote 9
N
by zuat.e
Source: Brazil MO 2013, problem #2
Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers known by both players, Arnaldo picks a number 
but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer 
(not necessarily in ). Then Arnaldo tells the number of divisors of 
. Show that Bernaldo can choose in a way that he can find out the number 
chosen by Arnaldo.
known by both players, Arnaldo picks a number 
but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer 
(not necessarily in ). Then Arnaldo tells the number of divisors of 
. Show that Bernaldo can choose in a way that he can find out the number 
chosen by Arnaldo.
9 replies
2025 TST 22
EthanWYX2009 1
N
by hukilau17
Source: 2025 TST 22
Let 
be a set of 2025 positive real numbers. For a subset 
, define 
as the median of 
when all elements of are arranged in increasing order, with the convention that 
. Define
![\[
P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.
\]](//latex.artofproblemsolving.com/1/d/f/1df2ea8487289111939a53dcc49576fbe6fdef37.png)
Find the smallest real number 
such that for any set of 2025 positive real numbers, the following inequality holds:
![\[
P(A) - Q(A) \leq C \cdot \max(A),
\]](//latex.artofproblemsolving.com/c/d/e/cdea343d11c722a423e857aeb01202863405773e.png)
where 
denotes the largest element in .
be a set of 2025 positive real numbers. For a subset 
, define 
as the median of 
when all elements of are arranged in increasing order, with the convention that 
. Define![\[
P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.
\]](http://latex.artofproblemsolving.com/1/d/f/1df2ea8487289111939a53dcc49576fbe6fdef37.png)
Find the smallest real number 
such that for any set of 2025 positive real numbers, the following inequality holds:![\[
P(A) - Q(A) \leq C \cdot \max(A),
\]](http://latex.artofproblemsolving.com/c/d/e/cdea343d11c722a423e857aeb01202863405773e.png)
where 
denotes the largest element in .
1 reply
Deriving Van der Waerden Theorem
Didier2 0
Source: Khamovniki 2023-2024 (group 10-1)
Suppose we have already proved that for any coloring of 
in 
colors, there exists an arithmetic progression of size 
. How can we derive Van der Waerden's theorem for 
from this?
in 
colors, there exists an arithmetic progression of size 
. How can we derive Van der Waerden's theorem for 
from this?
0 replies
Not so classic orthocenter problem
m4thbl3nd3r 6
N
by maths_enthusiast_0001
Source: own?
Let be circumcenter of a non-isosceles triangle and be a point in the interior of 
. Let 
be foots of perpendicular lines from to 
. Suppose that 
is cyclic and is the circumcenter of , 
. Prove that 
bisects
be circumcenter of a non-isosceles triangle and be a point in the interior of 
. Let 
be foots of perpendicular lines from to 
. Suppose that 
is cyclic and is the circumcenter of , 
. Prove that 
bisects
6 replies
Symmetric orthotransversal
G
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Source: Own
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Y by buratinogigle, Adventure10, Mango247
is incenter of and 
cut 
at 
Perpendicular to at cuts 
at 
respectively. Points 
and 
are defined cyclically. 
and 
are the orthotransversals of WRT and 
If
cuts at 
then by Desargues involution theorem for 
it follows that 
is an involution where is double and 
are symmetric WRT 
it coincides with the central symmetry WRT 
are symmetric WRT and similarly 
and 
are symmetric WRT and are symmetryc WRT 
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Y
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Y by buratinogigle, enhanced, Adventure10
My solution :
Let
be the incenter of 
and 
be its cevian triangle WRT .
Let the orthotransversal of WRT cuts 
at 
, respectively .
Let the orthotransversal of WRT cuts 
at 
, respectively .
Let
be the intersection of 
with A-external bisctor of ( it's well-known 
) .
From
and 
( both 
) 
.
Similarly, we can prove
and 
and 
are symmetry WRT .
Q.E.D
Let
be the incenter of 
and 
be its cevian triangle WRT .Let the orthotransversal of
WRT cuts 
at 
, respectively .Let the orthotransversal of
WRT cuts 
at 
, respectively .Let
be the intersection of 
with A-external bisctor of ( it's well-known 
) .From
and 
( both 
) 
.Similarly, we can prove
and 
and 
are symmetry WRT .Q.E.D
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Y
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Y by Adventure10, Mango247
Thank you so much, actually I have found solution the same as Telv base on idea of Luis but Telv posted. I only note that. Then line passes through I cuts 
at 
then 
are isogonal conjugate wrt 
.
at 
then 
are isogonal conjugate wrt 
.
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Y
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Y by buratinogigle, Adventure10, Mango247
Mr.Buratino wrote:
Prove that orthotransversal of incenter wrt reference triangle and wrt incentral triangle are symmetric through such this incenter (post #1).
Lemma 1. Let
be 2 points on the side of 
. We have: 
is tangent to 
are isogonal in 
Proof. Simple angle chasing.
Lemma 2. Given
. The line passing through which is tangent to 
intersects at . Prove that: 
Back to our main problem:
Consider
and the bisectors 
concur at The orthotransversal of
WRT intersects at 
The orthotransversal of
WRT 
intersects 
at 
In order to prove
, we only need to indicate that 
Notice that
are tangent to each other at (lemma 1)
is tangent to 
Now apply the lemma 2 for
we receive: and our proof is completed!Q.E.D
This post has been edited 1 time. Last edited by tranquanghuy7198, Jun 2, 2015, 4:50 PM
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