[/quote]
,
and 
.
,
is a subset of 
such that every element of 
.
![\[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \]](http://latex.artofproblemsolving.com/0/9/d/09d80fe5c25c5c6ea57c545b68c7e11b319cbada.png)
with integer coefficients 
,
,
,
and 
such that ![\[ f(a)=f(b)=f(c)=f(d)=5, \]](http://latex.artofproblemsolving.com/8/2/d/82d0c5d0c7a6dd48802ad8fbe04b211278c1fa58.png)
show that there is no integer 
such that 
.
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
such that 
divides by 
s
n
r
t
hats, numbered from 
hats. After that, each wise man (in turn: first the first, then the second, etc.) loudly calls a number, trying to guess the number of his hat. The numbers mentioned should not be repeated. When all the wise men have spoken, they take off their hats and check which one of them has guessed. Can the sages to act in such a way that at least three of them knowingly guessed?
be an acute triangle with circumcircle 
.
be the midpoint of 
and let 
be the midpoint of 
.
be the foot of the altitude from 
and let 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
be the incentre of acute-angled triangle 
,
,
respectively. Let line 
intersect the circumcircle of the triangle at 
and 
,
lies between 
and 
.
is internally tangent to 
at 
and 
be points on line 
is the midpoint of 
and 
is the midpoint of 
.
and 
meet at 
and intersect 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
such that points 
lie on the line 
is tangent to the segments 
and 
and to the circle 
is tangent to the segments 
and 
and to the circle 
and 
intersect at point 
.
be the set of all positive integers. Find all functions 
such that 
divides 
for all positive integers 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
are given. Losyash likes some of the cells of the 
checkerboard. In addition, he is interested in any checkered rectangle with a perimeter of 
,
is symmetric to the orthocenter 
.
.
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 
)
.
