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P lies on BC
Melid 0
Source: own
In scalene triangle 
, which doesn't have right angle, let 
be its circumcenter. Let 
and 
be orthocenters of triangle 
and 
, respectively. Let 
be circumcenter of triangle 
. If circle 
and circle 
intersect at 
for the second time, prove that lies on 
.
, which doesn't have right angle, let 
be its circumcenter. Let 
and 
be orthocenters of triangle 
and 
, respectively. Let 
be circumcenter of triangle 
. If circle 
and circle 
intersect at 
for the second time, prove that lies on 
.
0 replies
Polynomial functional equation
Fishheadtailbody 2
N
by Fishheadtailbody
Source: MACMO
is a polynomial with real coefficients such that![\[ P(x)^2 - 1 = 4 P(x^2 - 4x + 1). \]](http://latex.artofproblemsolving.com/b/4/2/b425a938a1dbbac25e0e0793b30c9e58845d45ab.png)
Find .2 replies
Strange circles in an orthocenter config
VideoCake 2
N
by pi_quadrat_sechstel
Source: 2025 German MO, Round 4, Grade 12, P3
Let 
and 
be altitudes in an acute triangle 
which meet at 
. Suppose that 
meets the circumcircle of at 
and 
such that lies on the shorter arc of 
and lies on the shorter arc of 
. Let 
and 
meet at 
. Show that the circumcircles of 
and 
and the line 
concur.
and 
be altitudes in an acute triangle 
which meet at 
. Suppose that 
meets the circumcircle of at 
and 
such that lies on the shorter arc of 
and lies on the shorter arc of 
. Let 
and 
meet at 
. Show that the circumcircles of 
and 
and the line 
concur.
2 replies
Lines pass through a common point
April 5
N
by SatisfiedMagma
Source: Baltic Way 2008, Problem 18
Let 
be a diameter of a circle 
, and let 
be the tangent at 
. Furthermore, let 
be a fixed, positive real, and consider all pairs of points 
and 
lying on , on opposite sides of , such that 
. The lines 
and 
intersect at points 
and 
, respectively. Show that all the lines 
pass through a common point.
be a diameter of a circle 
, and let 
be the tangent at 
. Furthermore, let 
be a fixed, positive real, and consider all pairs of points 
and 
lying on , on opposite sides of , such that 
. The lines 
and 
intersect at points 
and 
, respectively. Show that all the lines 
pass through a common point.
5 replies
Parameter and 4 variables
mihaig 1
N
by mihaig
Source: Own
Find the positive real constants 
such that
for all 
satisfying 
such that
for all 
satisfying 
1 reply
and 
.
How many friends can sit in that circle at most?
Arytva 0
A group of friends sits in a ring. Each friend picks a different whole number and holds a stone marked with it. Then they pass their stone one seat to the right so everyone ends up with two stones: one they made and one they received. Now they notice something odd: if your original number is
, your right-neighbor’s is 
, and the next person over is 
, then for every trio in the circle they see
They want as many friends as possible before this breaks (since all stones must stay distinct).
How many friends can sit in that circle at most?
0 replies
Reflected point lies on radical axis
Mahdi_Mashayekhi 7
N
by amogususususus
Source: Iran 2025 second round P4
Given is an acute and scalene triangle with circumcenter . 
and 
intersect the altitude from 
to at points and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
with circumcenter . 
and 
intersect the altitude from 
to at points and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
7 replies
Shortlist 2017/G3
fastlikearabbit 124
N
by ND_
Source: Shortlist 2017, Moldova TST 2018
Let be the circumcenter of an acute triangle . Line 
intersects the altitudes of through 
and 
at and , respectively. The altitudes meet at 
. Prove that the circumcenter of triangle 
lies on a median of triangle .
be the circumcenter of an acute triangle . Line 
intersects the altitudes of through 
and 
at and , respectively. The altitudes meet at 
. Prove that the circumcenter of triangle 
lies on a median of triangle .
124 replies
+1 w
