AoPS Academy
, 
be positive integers such that 
is a multiple of 
for all positive integers 
. Prove that 
.
with 
is called historic if 
. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
satisfying 
for all pairs of positive reals 
and 
. Here, 
denotes the set of all positive reals.
be positive real numbers such that : 
. Prove that : 
Note that 
.
is divisible by 
if and only if it is divisible by both 
and 
. If 
then we can take the numbers 
from distinct congruence classes modulo so that will not be divisible by . On the other hand, if 
then according to the pigeonhole principle there always exist 
and 
such that 
is divisible by (and of course there exist 
and 
such that 
is divisible by ).
the product of all differences 
is divisible by 
Therefore, 
.
and there are 
i
whose difference is 
,
integers work.
.
numbers from this set and the product of their differences cannot divide