AoPS Academy
be a prime and 
a nonempty set. Suppose that for each element 
there exist elements 
, 
such that 
is divisible by . Prove that 
, where 
denotes the cardinality of 
.
different point-like cities were founded one by one; from the second onwards, each newly founded city and the nearest existing city (the older one, if there were two) were declared sister cities. The surviving map of the empire shows the cities and the distances between them, but not the order in which they were founded. Historians have tried to deduce from the map that each city had at most 41 sister cities.
, give a map from which this deduction can be made.
, this conclusion cannot be drawn from any map.
such that:
,
is the incenter. Line 
intersects circumcircle of 
,
.
are points chosen on 
such that 
.
