be a function satisfying the following conditions for all 
:![\[
\begin{cases}
f(n+1) > f(n) \\
f(f(n)) = 2n + 2
\end{cases}
\]](http://latex.artofproblemsolving.com/0/4/7/047e28c9d899d7de443ae40ac2c756fda851fafb.png)
Find the value of 
.
has centroid 
.
passing through 
.
and 
.
is chosen on 
at 
and 
.
differents subjects, each consist of 
students. Given that for any 
different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least 
different subjects.
.
where 
and 
are integers and 
.
with real coefficients such that for all reals 
such that 
we have the following relations![\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]](http://latex.artofproblemsolving.com/4/4/9/4491a53f9c6f96c18e42eb984b8e83566140bfdb.png)
,
may have in the range 
, as 
.
be its incenter. A point 
intersects 
at 
,
intersects 
at 
.
intersects 
and 
,
and 
intersect at 
,
and 
intersect at 
.
. Prove that
,
is circumcircle. Let 
.
and 
,
(opposite to 
and tangents to 
)
is midpoint of the arc 
.
be a sequence of positive real numbers, and 
be a positive integer, such that![\[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\]](http://latex.artofproblemsolving.com/f/1/2/f12c9c5e0f154549118a33a2f359329333f432a7.png)
and ![\[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]](http://latex.artofproblemsolving.com/b/0/b/b0b2bd4507621302eeee27ae50ed9c0afd72c6ef.png)
Y by buratinogigle
Inside triangle 
, point 
is marked. Point 
is on segment 
, and point 
is on segment 
such that the circumcircles of triangles 
and 
are tangent to line 
. Lines are drawn through points 
and 
passing through the center of the circumcircle of triangle 
, and through points and passing through the center of the circumcircle of triangle 
. Prove that there exists a circle tangent to all four drawn lines.
, point 
is marked. Point 
is on segment 
, and point 
is on segment 
such that the circumcircles of triangles 
and 
are tangent to line 
. Lines are drawn through points 
and 
passing through the center of the circumcircle of triangle 
, and through points and passing through the center of the circumcircle of triangle 
. Prove that there exists a circle tangent to all four drawn lines.
be the centers of 
respectively (last circle exists from invert at 
so dw) we claim that the quadrilateral asked for has as incenter 
,
.![\[ \measuredangle O_2RC=\measuredangle PRC-\measuredangle PRO_2=\measuredangle APC+90+\measuredangle RQP=\measuredangle APC+90+\measuredangle RQB+\measuredangle BQP=\measuredangle BPC+90+\measuredangle ACB=\measuredangle BCO_1+\measuredangle ACB=\measuredangle RCO_1 \]](http://latex.artofproblemsolving.com/d/7/2/d72bd2b569e03b5fbc524303576c6f4721b28094.png)
And this shows that the perpendicular bisector of 
becomes an angle bisector so you do the same to get that so does 
and from here we are obviously done 
.
.
maps 
to 
,
are lie on a circle, assume centered at 
.
and 
,
intersects 
(other than 
)
and 
is an isosceles triangle, which implies 
.
