[/quote]
,
with 
,
to the sides 
are 
,
is the orthocenter. 
is the midpoint of segment 
.
and 
intersect at 
.
from 
and 
intersect at 
.
are colinear
be an integer and let 
be positive integers. In a group of 
people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:
.
,
games of chess.
.
is chosen from the minor arc 
such that 
.
be a point on 
is perpendicular to 
,
be a point on line 
is perpendicular to 
and 
,
is the reflection point of 
.
satisfying the following equations: 
such that: 
.
be positive real numbers such that 
.
.
be real numbers such that 
Prove that
Let 
Prove that
Let 
Prove that
.
be a real numbers such that 
P
Where 
P
Where 
I
be positive real numbers, show that:
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 
)
.
