[/quote]
,
such that
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 
,
.
is given. It turns out that for every positive integer number 
,
and 
are positive, they have the same number of divisors. Is it necessarily true that 
?
Prove that
such that
is self-bounded if for each 
,
there exist at least 
terms of the sequence that are less than or equal to 
for a self-bounded sequence 
such that 
is an integer.
cards on which the numbers 
,
,
,
be the sum of the numbers written on the cards that Bob keeps. Find the greatest positive integer 
for which Bob can guarantee that 
such that
,
.
, with a,b,c,d are arbitrary integers and p is an arbitrary prime such that the fraction is an integer too.
Y by
Let 
be a positive integer such that 
. Consider a grid with size 
where each square can be white or black, in the beginning they are all white. In every step we can change the colors of cells forming a shape like below
or any of its rotations. Determine all such that the whole grid can be black after a finite number of steps.
be a positive integer such that 
. Consider a grid with size 
where each square can be white or black, in the beginning they are all white. In every step we can change the colors of cells forming a shape like belowor any of its rotations. Determine all
such that the whole grid can be black after a finite number of steps.
