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.
, 
, 
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)
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 .
. How many complex numbers 
are there such that 
and 
is an integer.
then 
and 
. Note that 
and we can only have equality when 
. Then 
. This 
gives 120 solutions for , from now on 
.
then 
hence 
where 
. If 
then 
impossible. If 
then consider 
. We have 
and 
. This is impossible as we just saw.
gives no solutions. The final case is 
hence 
. This gives 600 solutions. Hence in total we have 720 solutions.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
.
and
Since this quantity is an integer, it must be real, so 
From here we have two cases.
(side note: is it an abuse of notation to use "mod" for non-integer values?), or, put simply, 
and 
are the same angle. In this case, 
is already 
,
possible values for 
or that 
Rearranging, we get that this is equivalent to 
which also means that 
so 
is an integer if and only if 
is. From here, we have several cases:
Hence, the only 
,
total possible 
different 
for 
.
such