[/quote]
,
and 
.
Prove 
Find maximum 
be reals such that 
and 
Prove that
Let 
Prove that
Prove that
and 
Prove that
Let 
Prove that
,
,
be the midpoints of sides BC, CA, and AB, respectively. Point 
is located on the ray 
so that 
is similar to 
.
and 
on rays 
and 
,
,
,
are concurrent, i.e. these three lines intersect at a point.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
be a triangle. Distinct points 
,
,
lie on sides 
,
,
,
and 
are cyclic. Line 
meets the circumcircle of 
again at 
.
denote the reflection of 
.
such that 
,
with 
is similar to 
.
.
be the set of real numbers. Find all functions 
that satisfy the following condition. Here, 
is the function obtained by composing 
times, that is, 
,
be an odd prime number and 
be the set of positive integers. Suppose that a function 
satisfies the following properties:
.
for any pair of relatively prime positive integers 
not both equal to 1;
for any pair of relatively prime positive integers 
.
be a point on 
such that 
.
the point on 
.
is the projection of 
passes through the orthocenter of 
Y by
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