[/quote]
,
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 an acute triangle. Let 
and 
such that lines 
and 
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 
to 
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.
and 
.
Equality holds when 
or 
.
at 
intersect again at 
.
Y by
Let 
be a triangle, 
its incenter and 
its circumcenter. Lines 
, 
and 
meet the circumcircle of at 
, 
and 
. The midpoints of 
, 
and 
are denoted by 
, 
and 
, respectively. Let 
be the the circumcenter of 
.
Prove that, and are collinear.
be a triangle, 
its incenter and 
its circumcenter. Lines 
, 
and 
meet the circumcircle of at 
, 
and 
. The midpoints of 
, 
and 
are denoted by 
, 
and 
, respectively. Let 
be the the circumcenter of 
.Prove that
, and are collinear.This post has been edited 3 times. Last edited by parmenides51, Sep 6, 2021, 8:54 PM
Using analogous perpendicularities we deduce that
Since 
and 
and 
.
,
.
which has the circumcircle as the nine point circle (say 
.
which has the circumcircle as the nine point circle (say 
