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 a acute triangle with 
. 
are the orthocenter and circumcenter. Let 
be a point on the same side of 
as 
, such that 
is equilateral. Let 
be a point on the same side of 
as , such that 
is equilateral. Let 
be a point on the same side of 
as , such that 
is equilateral. Let 
be the midpoint of 
. Prove that 
are collinear.
, show that![\[\left(1+\frac{a}{b}\right)\left(1+\frac{a}{b}+\frac{b}{c}\right)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\geq 16\]](http://latex.artofproblemsolving.com/c/1/7/c17cc42f0feafca5ecbeacde7076daa5b2615c60.png)
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
and a line 
in the plane.Let 
be reflections of 
across the line , respectively.Let 
be the midpoints of 
, respectively.Let 
be the reflections of across , respectively.Prove that the points lie on a line parallel to .
be the set of positive integers. Find all functions 
such that![\[f(m)+n \mid m^2f(m) - f(n) \]](http://latex.artofproblemsolving.com/8/5/5/85515a57bf60498134ceb168039fe734a5ff91a5.png)
for all positive integers 
and 
.
.
be an acute-angled triangle with 
, and let 
be its circumcentre and orthocentre respectively. Points 
lie on segments 
respectively, such that ![\[\angle ZOB=\angle YOC = 90^{\circ}.\]](http://latex.artofproblemsolving.com/8/f/e/8fe34cc0f04ff9a48f1a7eee89fc50d57272d936.png)
The perpendicular line from 
to line 
meets lines 
and 
at 
respectively. Let the tangents to the circumcircle of 
at points 
and 
meet at point 
. Prove that 
are concyclic.
marbles in 
colors such that there are exactly 
marbles of each color. She wants to arrange them on 
shelves, 
marbles on each shelf such that for any 
colors there is a shelf that has marbles of those colors.
be an inscribed circle of 
and touching 
, 
, 
at , 
, 
respectively. Let 
and 
be diameters of . Let 
and be the pole of 
and 
with respect to , respectively. 
cuts again at 
. 
cuts again at 
. The tangent at of cuts 
at . The tangent at of cuts 
at 
. Let 
and 
be midpoint of 
and 
, respectively.
, 
and perpendicular bisector of 
are concurrent.
such that 
.
and 
to be both integer, 's denominator must divide 
, which in particular means that 
is bounded below
is bounded above (by using the definition of 
for both and 
), which means that is bounded above too.
is unbounded, so from this and the boundedness of we immediately obtain that both sides are zero, so 
.
for some n, so this relation holds for all n, and ultimately the characterisation of sequences are all linear sequences with common difference k.
Prove that
Let 
Prove that
Prove that
Where 
satisfying
Prove
and 
Prove that
Where 
