be an odd prime number. How many 
of 
are there, the sum of whose elements is divisible by 
be a natural number. The numbers 
are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the 
GCDs obtained?
,
that makes the following equation have no integer root 
.
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
that satisfy the equation
Y by Adventure10, Mango247
Given a positive integer 
, let 
, 
, ..., 
denote 
subsets of a set 
such that each 
contains exactly two elements. Find the minimum value of 
such that for any such choice of subsets , , ..., , there exists a subset 
of such that:
(1)
;
(2)
for every 
.
, let 
, 
, ..., 
denote 
subsets of a set 
such that each 
contains exactly two elements. Find the minimum value of 
such that for any such choice of subsets , , ..., , there exists a subset 
of such that:(1)
;(2)
for every 
.
,
then not exist 
and 
for all 
.
max and 
so 
.
,
and 
satisfy 
( if not we can add 
in 
)
so 
.
different. So, 
satisfy 
for all 
and 
for all 
,we will get the result 
.
,
should imply 
.
.
.
.
so 
be two nonnegative integers such that 
.
for every 
and 
for every 
.
covers all of 
,
which does not lie in any of the 
itself collapses to a one-element set, and one-element sets can be ignored).
is indeed the minimum (for 
,
