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and separate them into two sets 
and 
of 10 numbers each. Suppose that 
are the numbers of , and 
are the numbers of . Determine all the possible values of the sum: 
. and are arranged in opposite order.
be a positive integer and 
a positive divisor of . Prove that if 
is a whole number, then it is equal to one.
, we define 
and 
as the set of positive divisors of of the forms 
and 
respectively. For example 
and 
. Show that for all , it is true that: 
denotes the quantity of elements in set 
.
be an odd positive integer. There is a graph 
with 
vertices such that if you partition the vertices into two groups and with vertices each, then the subgraph consisting of only vertices and edges within is connected and has a closed path containing all of its edges, starting and ending with the same vertex. The same condition is true for as well. Is necessarily a clique?
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
, 
, 
