D is incenter
by Layaliya, Apr 3, 2025, 11:03 AM
Given an acute triangle 
where 
. Point 
is the circumcenter of triangle , and 
is the projection of point 
onto line 
. The midpoints of , 
, and 
are 
, 
, and 
, respectively. The line 
intersects 
and 
at points 
and 
, respectively. Prove that is the incenter of triangle 
.
where 
. Point 
is the circumcenter of triangle , and 
is the projection of point 
onto line 
. The midpoints of , 
, and 
are 
, 
, and 
, respectively. The line 
intersects 
and 
at points 
and 
, respectively. Prove that is the incenter of triangle 
.
to 
(inclusive) are selected randomly without repetitions, then either all of them will be odd, or all will be divisible by 
,
such that 
.
and 
: Prove that :
be inscribed in the circumcircle 
and circumscribed about the incircle 
,
.
,
,
at 
,
,
,
,
,
,
,
,
(distinct from 
)
intersects 
(distinct from 
be the midpoint of the arc 
of 
and 
at 
and 
,
at 
.
.
make the equation ![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
and 
and 
respectively. The incenters of the triangles 
and 
are 
and 
respectively; the circumcenters of the triangles 
and 
are 
and 
respectively. Prove that 
and 
are parallel.
the product of all differences 
is divisible by 
.
April 2021
February 2021