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.
, define a function 
such that the inequality :![\[f(x) \leq f(x)^3-2f(x)^2+2\]](http://latex.artofproblemsolving.com/1/2/c/12c4c4c850c1c2886f002dfcb0bead075e2ab679.png)
.
of a sequence 
of positive integers satisfying both the following properties:
, and
(where 
) with a choice of signs 
for which ![\[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]](http://latex.artofproblemsolving.com/4/b/6/4b66fe235e9a56cc646dbd38aae72f8ddfeb77de.png)
[/list]
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
be an acute triangle with circumcenter 
and orthocenter 
, satisfying 
. The tangent line at 
to the circumcicle of intersects 
in 
. Let 
be the midpoint of 
. Prove that 
.
and 
be positive integers greater than 
. In each unit square of an 
grid lies a coin with its tail side up. A move consists of the following steps.
square in the grid;
for which it is possible that every coin shows head-side up after a finite number of moves.
be the incentre of acute-angled triangle . Let the incircle meet 
, and 
at 
, and 
respectively. Let line 
intersect the circumcircle of the triangle at 
and 
, such that 
lies between 
and . Prove that 
.
so that for every in 
.
such that 
. The lines 
and intersect at a point and the line passing through which is parallel to , intersects 
at . Prove that
be the circumcircle of acute triangle . Two tangents of from 
and 
intersect at , 
and intersect at 
. Point , are on 
and such that 
and 
.
are concyclic.
the centre of the circle passing through . 
, 
are difined similarly. Prove that 
, 
, 
are concurrent.
> 
that satisfy 
, for all 
€ .
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](http://latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
. A straight line touches the inner circle at 
and cuts the outer circle at 
and 
. Prove that angle 
angle 
.
be a line That touch the Out and the incircle at and Let 
be a point in and Let 
be the intersection of 
with The out circle , Then 
and 
and Thus 
in 
and in 
we have 
Which implies that 
and since 
and are cocyclique we get 
, hence the tangent to the external circle at the point of intersection of 
with it will be parallel to 
, done. is the external angle bisector of 
then.
and 
Prove that
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
