be integers such that, for each 
,
with![\[
\sum_{j \in S} a_j \equiv i \pmod{2048}.
\]](http://latex.artofproblemsolving.com/8/b/e/8be39aaf83795f611b1c3fa0d5971905bd6ebdfe.png)
Show that for each 
,
such that 
is divisible by 
but not by 
.
is the summation notation, for instance, 
,
,
.
be a triangle with incircle 
,
and 
at points 
,
on the line 
such that 
and 
.
and 
.
and 
as the intersections of 
and 
with lines 
and 
,
is the intersection of 
and 
.
be positive integers. Let 
,
,
be three roots of the polynomial 
.
,
,
,
,
,
,
.
and ![$(a, b) + [a, b]=2021^c$](http://latex.artofproblemsolving.com/2/4/6/246ccbb22acecb7adc72219afb8afb34c9b05101.png)
.
is a prime number, prove that the number 
is composite.
Y by Rounak_iitr
Given 
with circumcircle 
. Point 
in such that 
is the angle bisector of 
. Circle with center and radius 
intersects and 
at 
and 
respectively, 
. Let 
be the midpoint of arc in that didn't have 
. Prove that 
angle bisector of 
if and only if 
.
with circumcircle 
. Point 
in such that 
is the angle bisector of 
. Circle with center and radius 
intersects and 
at 
and 
respectively, 
. Let 
be the midpoint of arc in that didn't have 
. Prove that 
angle bisector of 
if and only if 
.
is cyclic. Then it's just simple angle chasing.
,
is a cyclic quadrilateral. Since 
and 
is the angle bisector of 
,
,
.
digit accuracy, but couldn't find the correct construction .-.
be 
.
)
is not isosceles, line 
if and only if 
.
