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.
have the same incenter 
.Prove that 
is concyclic iff 
is concurrent
+1 w
, 
, 
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 .
and be two circles, intersecting in two distinct points 
and 
. A circle 
touches the circles and externally in the points and 
.
and 
intersect on the line 
.
with any radius. and turn to straight lines 
and 
, that intersect on a point 
. Note that lies on . turn to a circle 
such that its tangent to the lines and , at the points 
and 
.
Thus is the angle bisector, and thus the other two angle bisectors will meet at the incenter of 
, which lies on the angle bisector .
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
.
.
.
.
Then, 
and 
so 
The Angle Bisector Theorem finishes. 