IMO 2009 P2, but in space
by Miquel-point, May 14, 2025, 6:35 PM
Let 
be a tetrahedron with circumcenter 
. Suppose that the points 
and 
are interior points of the edges 
and 
, respectively. Let 
and 
be the centroids of the triangles 
, 
and 
, respectively. Prove that if the plane is tangent to the sphere 
then 
be a tetrahedron with circumcenter 
. Suppose that the points 
and 
are interior points of the edges 
and 
, respectively. Let 
and 
be the centroids of the triangles 
, 
and 
, respectively. Prove that if the plane is tangent to the sphere 
then 
D1031 : A general result on polynomial 1
by Dattier, May 14, 2025, 5:14 PM
Let 
with 
.
Is it true that![$P(x,y) \in \mathbb Q[x,y]$](//latex.artofproblemsolving.com/b/3/7/b37942b05292a493c8e350322adc122205bc778d.png)
?
with 
.Is it true that
![$P(x,y) \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/b/3/7/b37942b05292a493c8e350322adc122205bc778d.png)
?Anything real in this system must be integer
by Assassino9931, May 9, 2025, 9:26 AM
Determine the largest integer 
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
Marek Maruin, Slovakia
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.Marek Maruin, Slovakia
This post has been edited 1 time. Last edited by Assassino9931, May 9, 2025, 9:26 AM
Old hard problem
by ItzsleepyXD, Apr 25, 2025, 4:15 AM
Let 
be a triangle and let be its circumcenter and 
its incenter.
Let
be the radical center of its three mixtilinears and let 
be the isogonal conjugate of .
Let
be the Gergonne point of the triangle .
Prove that line
is parallel with line 
.
be a triangle and let be its circumcenter and 
its incenter.Let
be the radical center of its three mixtilinears and let 
be the isogonal conjugate of .Let
be the Gergonne point of the triangle .Prove that line
is parallel with line 
.Easy Geometry
by pokmui9909, Mar 30, 2025, 5:18 AM
Triangle satisfies 
. Let the incenter of triangle be 
, which touches 
at 
, respectively. Let 
be the midpoint of 
. Let the circle centered at passing through 
intersect 
at 
, respecively. Let line 
meet at , line 
meet 
at 
. Prove that the three lines 
are concurrent.
satisfies 
. Let the incenter of triangle be 
, which touches 
at 
, respectively. Let 
be the midpoint of 
. Let the circle centered at passing through 
intersect 
at 
, respecively. Let line 
meet at , line 
meet 
at 
. Prove that the three lines 
are concurrent.This post has been edited 1 time. Last edited by pokmui9909, Mar 30, 2025, 5:29 AM
such that
for all 
.\frac{2^{n!}-1}{2^n-1} be a square
by AlperenINAN, May 13, 2024, 8:49 AM
Find all positive integer values of 
such that the value of the
is a square of an integer.
such that the value of the
is a square of an integer.This post has been edited 1 time. Last edited by AlperenINAN, May 13, 2024, 9:26 AM
Chinese Girls Mathematical Olympiad 2017, Problem 7
by Hermitianism, Aug 16, 2017, 9:29 AM
This is a very classical problem.
Let the be a cyclic quadrilateral with circumcircle 
.Lines 
and 
intersect at point 
,and lines , intersect at point 
.Circle 
is tangent to segments 
at points 
respectively,and intersects with circle at points 
.Lines 
intersect line 
at 
respectively.Show that 
are concyclic.
Let the
be a cyclic quadrilateral with circumcircle 
.Lines 
and 
intersect at point 
,and lines , intersect at point 
.Circle 
is tangent to segments 
at points 
respectively,and intersects with circle at points 
.Lines 
intersect line 
at 
respectively.Show that 
are concyclic.Attachments:
This post has been edited 1 time. Last edited by Hermitianism, Aug 17, 2017, 9:04 AM
CIIM 2011 First day problem 3
by Ozc, Oct 3, 2014, 2:32 AM
Let 
be a rational function with complex coefficients whose denominator does not have multiple roots. Let 
be the complex roots of 
and 
be the roots of 
. Suppose that 
is a simple root of . Prove that
![\[ \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.\]](//latex.artofproblemsolving.com/6/f/3/6f36b138d97a24d1941d21f7ffbc11530ca2adde.png)
be a rational function with complex coefficients whose denominator does not have multiple roots. Let 
be the complex roots of 
and 
be the roots of 
. Suppose that 
is a simple root of . Prove that![\[ \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.\]](http://latex.artofproblemsolving.com/6/f/3/6f36b138d97a24d1941d21f7ffbc11530ca2adde.png)
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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