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such that 
for all 
?
be an inscribed quadrilateral. Circles of diameters 
and 
intersect at points 
and 
, and circles of diameters 
and 
intersect at points 
and 
. The circles of diameters 
and 
intersect in two points 
and 
. Prove that the lines 
and 
are concurrent.
such that 
is a natural number.
.
such that 
.
be a positive real number. Find all functions 
such that 
and![\[
f(af(x) - f(y) + y) \leq x + f(y) - y, \quad \forall x, y \in \mathbb{R}^+.
\]](http://latex.artofproblemsolving.com/e/2/9/e296b71ad8b8bd977e2bb2a3630a5bc020b8e03a.png)
and 
be the incircle and circumcircle of the acute triangle 
, respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on 
. Extend 
and 
, intersecting at 
and 
, respectively. Prove that 
and 
intersects on .
and 
Prove that
Let 
and 
Prove that
. Find all functions 
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
such that 
and 
for any real numbers 
and 
.
is the only function satisfies the given conditions.
and 
, 
,
, 
as 
as but 
and 
hold is if 
, 
, 
Now 
for all 
which clearly works.
be an increasing sequence of positive integer such that 
.
there exists 
and 
.
for all 
in both, we find 
and 
,
in the second, we find 
.
.
.
for all 
,
,
too. Thus, 
,
,
,
,