AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
and 
be positive integers. Suppose that the sum of integers between and , including and , are equal to 
.
, find the pair such that achieves the minimum.
be real numbers such that 
and
be integers such that, for each 
, there exists a subset 
with![\[
\sum_{j \in S} a_j \equiv i \pmod{2048}.
\]](http://latex.artofproblemsolving.com/8/b/e/8be39aaf83795f611b1c3fa0d5971905bd6ebdfe.png)
Show that for each 
, there is exactly one 
such that 
is divisible by 
but not by 
.
is the summation notation, for instance, 
, while for the empty set 
, one defines 
.
be an acute-angled triangle with 
. Let 
be the circumcircle of . Let 
be the midpoint of the arc 
of containing 
. The perpendicular from to 
meets 
at 
and meets again at 
. The line through parallel to meets line 
at 
. Denote the circumcircle of triangle 
by 
. Let meet again at 
. Prove that the line tangent to at 
meets line on the internal angle bisector of 
.
be an acute-angled triangle with 
. The circle with diameter intersects the sides 
and 
at 
and 
respectively. Denote by 
the midpoint of the side . The bisectors of the angles and 
intersect at 
. Prove that the circumcircles of the triangles 
and 
have a common point lying on the side .
. Given an infinite sequence of positive real numbers 
, Turbo successively crawls distances around the circle, each time choosing to crawl either clockwise or counterclockwise.
with the following property: for every sequence of positive real numbers with 
for all 
, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.
is inscribed in circle . Let 
denote its incenter and 
its -excenter. Let denote the midpoint of arc 
. Line 
meets a second time at 
. The perpendicular to 
at meets sides and at 
and 
respectively. The circumcircle of 
meets 
a second time at , and the circumcircle of 
meets 
a second time at 
. Prove that 
passes through the antipodal to on .
, prove that![\[
2+ \frac{B+C-A}{\sin A} +\frac{C+A-B}{\sin B} +\frac{A+B-C}{\sin C}
> \frac{\pi}{2}\left( \frac{\sin^2 A+\sin^2 B+\sin^2 C}{\sin A \sin B \sin C } \right)
\]](http://latex.artofproblemsolving.com/2/0/1/2012bbef4475a5c433a4e54c53195c729c5bab99.png)
. Let be a subset of such that no sum of three distinct elements of is divisible by 
. What is the maximum value of 
.
, we can include elements that are congruent to 
and 
, that is, numbers of the form 
and 
. is 40.
satisfying
Prove
elements such that the sum would be divisible by 
(mod 5) are compatible with each other in terms of the constraints. In fact, this is the only residue duo that satisfies the constraints given for A. Thus, it maximizes the amount of elements.
elements in 