of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of 
be the reflection of 
Prove that the circles 
and 
are tangent.
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x^2 + y) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/c/7/c/c7cd7c9a4d4372afc23e4503bb1c05fc7d07e788.png)
be positive real numbers satisfying![\[
a^2 + c^2 = b(a + c).
\]](http://latex.artofproblemsolving.com/9/c/0/9c0730d15b1177ee23d3f65bc2fe8b2cc85d0c62.png)
Let![\[
m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right).
\]](http://latex.artofproblemsolving.com/e/b/6/eb6f340defa4b38e35292062b202d138b9de843b.png)
Find the value of 
.
Y by farhad.fritl, cubres
An infinite increasing sequence 
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to . Show that there exists an infinite sequence 
of positive integers such that for every central sequence 
, there are infinitely many positive integers with 
.
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to . Show that there exists an infinite sequence 
of positive integers such that for every central sequence 
, there are infinitely many positive integers with 
.
