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Circumcircle of MUV tangent to two circles at once
MathMystic33 1
N
by ariopro1387
Source: Macedonian Mathematical Olympiad 2025 Problem 1
Given is an acute triangle 
with 
. Let 
be the midpoint of side 
, and let 
and 
be points on segments 
and 
, respectively, such that 
. Let 
be the circumcircle of 
, and 
the circumcircle of 
. The common tangent 
to and , which lies closer to point 
, touches and at points 
and 
, respectively. Let the line 
intersect again at 
, and the line 
intersect again at 
. Prove that the circumcircle of triangle 
is tangent to both and .
with 
. Let 
be the midpoint of side 
, and let 
and 
be points on segments 
and 
, respectively, such that 
. Let 
be the circumcircle of 
, and 
the circumcircle of 
. The common tangent 
to and , which lies closer to point 
, touches and at points 
and 
, respectively. Let the line 
intersect again at 
, and the line 
intersect again at 
. Prove that the circumcircle of triangle 
is tangent to both and .1 reply
Concurrency of tangent touchpoint lines on thales circles
MathMystic33 0
Source: 2024 Macedonian Team Selection Test P4
Let 
be an acute scalene triangle. Denote by 
the circle with diameter 
, and let 
be the contact points of the tangents from 
to , chosen so that 
and 
lie on opposite sides of 
and 
and 
lie on opposite sides of 
. Similarly, let 
be the circle with diameter 
, with tangents from touching at 
, and 
the circle with diameter , with tangents from touching at 
.
Prove that the lines
are concurrent.
be an acute scalene triangle. Denote by 
the circle with diameter 
, and let 
be the contact points of the tangents from 
to , chosen so that 
and 
lie on opposite sides of 
and 
and 
lie on opposite sides of 
. Similarly, let 
be the circle with diameter 
, with tangents from touching at 
, and 
the circle with diameter , with tangents from touching at 
.Prove that the lines
are concurrent.
0 replies
Equal areas of the triangles on the parabola
NO_SQUARES 0
Source: Regional Stage of ARO 2025 10.10; also Kvant 2025 no. 3 M2837
On the graphic of the function 
were selected 
pairwise distinct points, abscissas of which are integer numbers from the segment ![$[0; 100000]$](//latex.artofproblemsolving.com/0/5/e/05e1d2855ef6a6dfaf9b24f2713455ab1017a30d.png)
. Prove that it is possible to choose six different selected points , , , 
, 
, 
such that areas of triangles 
and 
are equals.
A. Tereshin
were selected 
pairwise distinct points, abscissas of which are integer numbers from the segment ![$[0; 100000]$](http://latex.artofproblemsolving.com/0/5/e/05e1d2855ef6a6dfaf9b24f2713455ab1017a30d.png)
. Prove that it is possible to choose six different selected points , , , 
, 
, 
such that areas of triangles 
and 
are equals.A. Tereshin
0 replies
Concurrency from symmetric points on the sides of a triangle
MathMystic33 0
Source: 2024 Macedonian Team Selection Test P3
Let be a triangle. On side take points 
and 
such that 
on side take points 
and 
such that 
and on side take points 
and 
such that 
Let 
and 
Prove that the lines
are concurrent.
be a triangle. On side take points 
and 
such that 
on side
take points 
and 
such that 
and on side take points 
and 
such that 
Let 
and 
Prove that the lines
are concurrent.
0 replies
Grouping angles in a pentagon with bisectors
Assassino9931 2
N
by Assassino9931
Source: Al-Khwarizmi International Junior Olympiad 2025 P2
Let 
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](//latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
.
Miroslav Marinov, Bulgaria
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
.Miroslav Marinov, Bulgaria
2 replies
Geometric inequality with 2 orthocenters and midpoint of the side
NO_SQUARES 0
Source: Regional Stage of ARO 2025 10.5; also Kvant 2025 no. 3 M2836
The heights 
and 
of the acute-angled triangle intersect at point 
, the heights of the triangle 
intersect at point 
, point is the midpoint of side . Prove that 
.
A. Kuznetsov
and 
of the acute-angled triangle intersect at point 
, the heights of the triangle 
intersect at point 
, point is the midpoint of side . Prove that 
.A. Kuznetsov
0 replies
Taking antipode on isosceles triangle's circumcenter
Nuran2010 1
N
by Sadigly
Source: Azerbaijan Al-Khwarizmi IJMO TST 2025
In isosceles triangle, the condition 
is satisfied. Point 
is taken on the circumcircle of such that 
.A line parallel to which passes from intersects and respectively at 
and .Show that circumcircle of passes from circumcenter of 
.
is satisfied. Point 
is taken on the circumcircle of such that 
.A line parallel to which passes from intersects and respectively at 
and .Show that circumcircle of passes from circumcenter of 
.
1 reply
Proving ZA=ZB
nAalniaOMliO 7
N
by nAalniaOMliO
Source: Belarusian National Olympiad 2025
Point is the foot of the altitude from of triangle . On the lines and points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.
Prove that
.
Vadzim Kamianetski
is the foot of the altitude from of triangle . On the lines and points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.Prove that
.Vadzim Kamianetski
7 replies
Tangents involving a centroid with an isosceles triangle result
pithon_with_an_i 2
N
by Funcshun840
Source: Revenge JOM 2025 Problem 5, Revenge JOMSL 2025 G5, Own
A triangle has centroid 
. A line parallel to passing through intersects the circumcircle of at a point . Let lines 
and intersect at . Suppose a point is chosen on such that the tangent of the circumcircle of 
at , the tangent of the circumcircle of at and concur. Prove that 
.
has centroid 
. A line parallel to passing through intersects the circumcircle of at a point . Let lines 
and intersect at . Suppose a point is chosen on such that the tangent of the circumcircle of 
at , the tangent of the circumcircle of at and concur. Prove that 
.2 replies
orthocenter on sus circle
DVDTSB 1
N
by Double07
Source: Romania TST 2025 Day 2 P1
Let 
be an acute triangle with , and let 
be the center of its circumcircle. Let 
be the reflection of with respect to . The line through parallel to intersects 
at 
, and the tangent at to the circle 
intersects the line through parallel to at point . Let 
be a point on the ray 
, starting at , such that 
.
Show that the orthocenter of triangle lies on the circle with diameter 
.
Proposed by Radu Lecoiu
be an acute triangle with , and let 
be the center of its circumcircle. Let 
be the reflection of with respect to . The line through parallel to intersects 
at 
, and the tangent at to the circle 
intersects the line through parallel to at point . Let 
be a point on the ray 
, starting at , such that 
.Show that the orthocenter of triangle
lies on the circle with diameter 
.Proposed by Radu Lecoiu
1 reply
