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such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
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)
be a regular heptagon with its center 
. 
is the orthocenter of triangle 
, 
is the incenter of triangle 
. Let 
be the midpoint of 
and 
be the intersection point of 
and 
. Assume 
is the circumcenter of triangle 
. Prove that 
concur at a single point.
be a positive integer. We say that is bao-good if there exist integers 
such that:
for which![\[
a \mid s^b - l.
\]](http://latex.artofproblemsolving.com/5/d/9/5d9db7678c54d1c6a909b4aca39a329cadd71e2c.png)
2. For every proper divisor 
of (that is, 
and 
), there exists a positive integer such that![\[
a' \mid s^b - l.
\]](http://latex.artofproblemsolving.com/4/b/2/4b26b9bd3da8958287703c3f575bf5d5da7fd477.png)
Determine all bao-good positive integers .
be the incircle and a circumcenter of triangle 
such that 
. On sides 
and 
there are points 
and 
, respectively, such that 
. Prove that 
and 
