[/quote]
,
be the circumcenter, symmedian point of 
.
at 
,
at 
,
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
be a point on side 
and 
be a point on side 
such that lines 
and 
are parallel. Let 
be an interior point of 
.
and 
meet side 
and 
,
and 
intersect at a point 
.
,
are collinear.
.
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
is given. Let 
be the intersection point of 
lies on the arc 
.
.
be a trapezium inscribed in a circle 
with diameter 
. The circle with center 
meets 
and 
(where 
at 
.
is perpendicular to 
.
![\[ x + y^2 = p^m \]](http://latex.artofproblemsolving.com/e/c/f/ecf553036f22327dd457dc06611b6bc1b0ef072c.png)
![\[ x^2 + y = p^n \]](http://latex.artofproblemsolving.com/5/b/d/5bd672934a395aabd6865ca4b50d8da75c0fe07e.png)
For 
positive integers and 
being a prime.
Y by Adventure10, Mango247
Let 
be an acute angled triangle with orthocenter 
and circumcenter 
. Assume the circumcenter 
of 
lies on the circumcircle of 
. Reflect 
across to obtain 
, and let the lines 
and 
meet at 
. Let 
and 
be the midpoints of ![$\left[ XB \right],\left[ XC \right]$](//latex.artofproblemsolving.com/6/9/4/694f12c592e893e0f7d518b10e22769db6821ab1.png)
and ![$\left[ BC \right]$](//latex.artofproblemsolving.com/4/c/1/4c1df198b314e42d7d486ef5f31835dd1a4446a9.png)
, respectively. Prove that the points 
and 
are concyclic.
be an acute angled triangle with orthocenter 
and circumcenter 
. Assume the circumcenter 
of 
lies on the circumcircle of 
. Reflect 
across to obtain 
, and let the lines 
and 
meet at 
. Let 
and 
be the midpoints of ![$\left[ XB \right],\left[ XC \right]$](http://latex.artofproblemsolving.com/6/9/4/694f12c592e893e0f7d518b10e22769db6821ab1.png)
and ![$\left[ BC \right]$](http://latex.artofproblemsolving.com/4/c/1/4c1df198b314e42d7d486ef5f31835dd1a4446a9.png)
, respectively. Prove that the points 
and 
are concyclic.This post has been edited 2 times. Last edited by parmenides51, May 28, 2020, 5:41 AM
Reason: typos
Reason: typos
and 
,
,
is the incentre of 
thus 
,
by the radical axis again. Therefore, we have 
where the first equality is well-known and since 
,
being a parallelogram, therefore 
.
and we obtain that 
.
is an equaliteral triangle. 
and 
because of 
triangle.
are cyclic and 
