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, 
,
,
.
,
and 
on the line 
such that 
and 
.
be the intersection of lines 
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 Mango247, Mango247, Mango247
Let 
be a triangle, 
the center of the segment 
and 
a point on the straight line 
so that 
is between 
and . Let 
be a point on the other side of the line 
as , so that 
and 
. Let 
be the intersection of 
with the parallel to 
passing through and let 
be the intersection of 
with 
. Show that the lines 
and 
are concurrent.
be a triangle, 
the center of the segment 
and 
a point on the straight line 
so that 
is between 
and . Let 
be a point on the other side of the line 
as , so that 
and 
. Let 
be the intersection of 
with the parallel to 
passing through and let 
be the intersection of 
with 
. Show that the lines 
and 
are concurrent.This post has been edited 1 time. Last edited by parmenides51, Dec 24, 2020, 11:30 AM
