[/quote]
,
in primes 
.
angle 
is greater tha
.
is midpoint of 
.
and 
are the feet of the altitude from 
and 
are midpoint of 
and 
respectively. If 
intersect the line through 
parallel to 
,
.
,
be positive real numbers, such that 
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
and circumcircle 
.
and 
at 
,
intersect the circumcircle of triangle 
again at 
.
at 
and 
intersect on 
numbers on circle, and no one number is divided by other. In same time for all numbers we make next operation:
are two neighbors 
number 
and erase 
we can receive ?
numbers 
then after operation we receive circle with numbers 
.
.![\[
\sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2}
\]](http://latex.artofproblemsolving.com/9/0/3/903220c471bdd3028dda2fcb13874a4bcf752872.png)
without using the Rearrangement Inequality or Chebyshev's Inequality.
?
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
are non-negative real numbers and 
Prove that
the equality hold for 
or 
or any
be positive real numbers. Prove 
Prove that
,
.
.
such that for all positive real numbers 
and 
:
Y by
Let 
be a positive integer and 
be a family of at most 
distinct subsets of the set 
with the following property: we can consider distinct points in the plane, labelled 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to subsets in for any 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
be a positive integer and 
be a family of at most 
distinct subsets of the set 
with the following property: we can consider distinct points in the plane, labelled 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to subsets in for any 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
if and only if there is segment between 
.
be a number of subsets, which contain 
.
for some integers 
and 
.
from our first sentence. This means that all 
,
have to pairwise different. This means that 
,
,
for subsets 
(because 
)
.
