[/quote]
,
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 the circumcircle of 
be the midpoint of the arc 
of 
.
at 
.
at 
.
by 
.
.
meets line 
.
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
.
and 
respectively. Denote by 
the midpoint of the side 
intersect at 
.
and 
have a common point lying on the side 
,
,
be real positive numbers. Show that ![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]](http://latex.artofproblemsolving.com/0/9/5/0959464bbb9b821a11df292f9e6addded7326371.png)
.
,
with the following property: for every sequence of positive real numbers 
for all 
,
is inscribed in circle 
denote its incenter and 
its 
.
meets 
.
at 
meets 
a second time at 
meets 
a second time at 
.
passes through the antipodal to 
,![\[
2+ \frac{B+C-A}{\sin A} +\frac{C+A-B}{\sin B} +\frac{A+B-C}{\sin C}
> \frac{\pi}{2}\left( \frac{\sin^2 A+\sin^2 B+\sin^2 C}{\sin A \sin B \sin C } \right)
\]](http://latex.artofproblemsolving.com/2/0/1/2012bbef4475a5c433a4e54c53195c729c5bab99.png)
satisfying
Prove
is 
than the set of numbers 
if among all inequalities of the form 
the number of true inequalities is at least 
times greater than the number of false ones. Prove that there do not exist three sets 
such that 
is 
,
,
is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
the ratio 
is an integer.
and 
.
?
so that the points 
are separated by 
and 
.
is tangent to the segments 
,
is tangent to the segments 
,
and 
meet at 
.
such that 
is an integer.
Y by
Let 
be an isosceles triangle with 
and 
. Let 
be the midpoint of 
and let 
be a point in 
such that 
is perpendicular to . The line parallel to passing through intersects side 
at point 
.If 
, determine the length of the segment 
.
be an isosceles triangle with 
and 
. Let 
be the midpoint of 
and let 
be a point in 
such that 
is perpendicular to . The line parallel to passing through intersects side 
at point 
.If 
, determine the length of the segment 
.This post has been edited 2 times. Last edited by parmenides51, Dec 5, 2021, 7:28 AM
.
and 
.
.
.
