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Regional, national, and international math olympiads
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Topic
First Poster
Last Poster
Sharygin 2025 CR P7
Gengar_in_Galar 5
N
an hour ago
by Blackbeam999
Source: Sharygin 2025
Let
,
be the incenter and the
-excenter of a triangle
;
,
be the touching points of the incircle with
,
respectively;
be the common point of
and
. The perpendicular to
from
meets
at point
. Prove that
,
,
,
are concyclic.
Proposed by:Y.Shcherbatov



















Proposed by:Y.Shcherbatov
5 replies

60^o angle wanted, equilateral on a square
parmenides51 5
N
an hour ago
by sunken rock
Source: 2019 Austrian Mathematical Olympiad Junior Regional Competition , Problem 2
A square
is given. Over the side
draw an equilateral triangle
on the outside. The midpoint of the segment
is
and the midpoint of the side
is
. Prove that
.
.
(Karl Czakler)








.
(Karl Czakler)
5 replies
Random walk
EthanWYX2009 0
an hour ago
As shown in the graph, an ant starts from
and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching
without passing through



0 replies
Lemma on tangency involving a parallelogram with orthocenter
Gimbrint 0
2 hours ago
Source: Own
Let
be an acute triangle (
) with circumcircle
and orthocenter
. Let
be the midpoint of
. Line
intersects
again at
, and line
intersects
again at
. Points
and
lie on
and
respectively, such that
is a parallelogram.
Prove that
is tangent to the circumcircle of triangle
.

















Prove that


0 replies



Consecutive squares are floors
ICE_CNME_4 10
N
2 hours ago
by JARP091
Determine how many positive integers

![\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]](http://latex.artofproblemsolving.com/d/e/c/dec527c4673470e7023d6aed00fac90268bd2fcc.png)
10 replies

Sharygin 2025 CR P10
Gengar_in_Galar 2
N
2 hours ago
by Kappa_Beta_725
Source: Sharygin 2025
An acute-angled triangle with one side equal to the altitude from the opposite vertex is cut from paper. Construct a point inside this triangle such that the square of the distance from it to one of the vertices equals the sum of the squares of distances to to the remaining two vertices. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines.
Proposed by: M.Evdokimov
Proposed by: M.Evdokimov
2 replies
Sharygin 2025 CR P12
Gengar_in_Galar 8
N
2 hours ago
by Kappa_Beta_725
Source: Sharygin 2025
Circles
and
are given. Let
be the midpoint of the segment joining their centers,
,
be arbitrary points on
,
respectively such that
. Find the locus of the midpoints of segments
.
Proposed by: L Shatunov









Proposed by: L Shatunov
8 replies
Sharygin 2025 CR P17
Gengar_in_Galar 6
N
3 hours ago
by Kappa_Beta_725
Source: Sharygin 2025
Let
,
be the circumcenter and the incenter of an acute-angled scalene triangle
;
,
,
be the touching points of its excircle with the side
and the extensions of
,
respectively. Prove that if the orthocenter of the triangle
lies on the circumcircle of
, then it is symmetric to the midpoint of the arc
with respect to
.
Proposed by: P.Puchkov,E.Utkin













Proposed by: P.Puchkov,E.Utkin
6 replies
Sharygin 2025 CR P21
Gengar_in_Galar 4
N
3 hours ago
by Kappa_Beta_725
Source: Sharygin 2025
Let
be a point inside a quadrilateral
such that
. Points
,
,
are isogonally conjugated to
with respect to the triangles
,
,
,
respectively. Prove that the diagonals of the quadrilaterals
and
concur.
Proposed by: G.Galyapin














Proposed by: G.Galyapin
4 replies
Sharygin 2025 CR P18
Gengar_in_Galar 6
N
3 hours ago
by Kappa_Beta_725
Source: Sharygin 2025
Let
be a quadrilateral such that the excircles
and
of triangles
and
touching their sides
and
respectively touch the extension of
at the same point
. The segment
meets
at point
, and the line
meets
at
and
. Prove that one of angles
and
is right
Proposed by: I.Kukharchuk


















Proposed by: I.Kukharchuk
6 replies
