AoPS Academy
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
, there will be a colour coloring exactly one integer from the set 
.
is a permutation of the elements of the set {
}?
and 
be different prime numbers. We are given an infinite decreasing arithmetic progression in which each of the numbers 
and 
occurs. Show that the numbers and also occur in this progression.
and be two prime numbers, with being a divisor of 
. Prove that there exist integers 
such that the polynomial 
is divisible by the polynomial 
with 
is a multiple of and 
.
