AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
and 
we define:
A 
between and as a sequence of distinct cities 
, 
, such that there are direct flights between 
and 
for every 
;
A 
between and as a path between and such that no other path between and has more cities;
A 
between and as a path between and such that no other path between and has fewer cities. and in the country, there exist a long path and a short path between them that have no cities in common (except and ). Let 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of , find all possible values 
such that 
for all real number 
and 
be a sequence of perfect squares such that 
can be obtained by concatenating a digit to the right of 
. Determine all such sequences that are of maximum length.
and 
.
?
be unit vectors on a plane, where is an odd number. Prove that there exist 
for 
such that 
st for every two subsets 
of 
, there exists a polynomial 
with integer coefficients st either 
or 
where the equations are considered mod n.
is the set of reminders of 
where 
mod n.
be a positive integer such that 
.
.
be Rutthee sequence if 
be a positive integer and
for all 
and there are some positive integer sastisfied 
such that 
and 
also find all possible value of in Rutthee sequence
be triangle with point 
and on 
and 
are on the same side of 
be midpoint of segment and 
be midpoint of segment 
be intersection of 
with 
and 
be two positive integers, such that 
. Find all the integer values that 
can take, where ![\[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]](http://latex.artofproblemsolving.com/c/b/2/cb27ccbcde6d90f98e10f8f6876c36ae8972af7f.png)
be the maximum of sums of three elements of 
taken over all and 
be the minimum correspondence. Define 
. The maximum value 
for 
is 
. Meanwhile, 
Conclude the rest using pigeonhole.
. Prove that
Prove that
Let 
Prove that
be a subset of 
such that 
.
of 
and the sum of all elements in 
is equal to the sum of all elements in 
.