[/quote]
,
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
,
and orthocenter 
.
is constant.
such that![\[
f(f(x)+y) = x+f(f(y))
\]](http://latex.artofproblemsolving.com/9/d/7/9d780faa5901646f7035e88400a9e3d8023f04ff.png)
for all 
.
points in a circle and number them arbitrarily with numbers from 
to 
pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most 
.
satisfies that for any positive integer 
and 
Suppose there is 
non-negative reals 
such that for 
Determine the least possible value of 
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
into a number 
.
and 
Prove that
and 
on 
and 
be such that 
and 
.
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
,
and 
are concurrent.
rows and 
columns. Alice and Eva take turns filling the empty cells of the first row of the table, Alice plays first. In each move, Alice chooses an empty cell and puts a cross in it; Eva in each move chooses an empty cell and puts a zero. When no empty cells remain in the first row, the players move on to the second row, and so on (in each new row Alice plays first).
so that:
with all real numbers 
satisfies 
and 
Prove that
.
Let 
.
Y by
On the sides 
and 
of the triangle 
, points 
and 
are marked, respectively, such that 
. Let 
be the intersection point of the segments 
and 
. Prove that the center of the inscribed circle of triangle lies on the circumcircle of triangle 
.
and 
of the triangle 
, points 
and 
are marked, respectively, such that 
. Let 
be the intersection point of the segments 
and 
. Prove that the center of the inscribed circle of triangle lies on the circumcircle of triangle 
.
or 
then 
which is absurd
then implicating it is parallel
by ceva
implicating BK is also bisector thus Q is incenter
