be an integer. Let 
and 
be polynomials with real coefficients such that the points 
in 
are the vertices of a regular 
-
,
statements are equivalent:
is differentiable, with continuous first derivative.
and for any two sequences 
,
,
for any positive integer 
,
is convergent.
Y by
Consider the set 
of all five-digit natural numbers whose decimal representation is obtained by permuting the digits 1, 2, 3, 4, 5. Prove that can be partitioned into two subsets such that the sum of the squares of the numbers in each of them is the same.
of all five-digit natural numbers whose decimal representation is obtained by permuting the digits 1, 2, 3, 4, 5. Prove that can be partitioned into two subsets such that the sum of the squares of the numbers in each of them is the same.
equivalence classes of size 
by considering the relation 
iff 
'
.
.
because 
and 
in the same place. Moreover, it is clear that 
.![$[24]$](http://latex.artofproblemsolving.com/b/6/8/b6871e37250e56779e9b365ea7b097aedfff3189.png)
.
so that 
iff 
or there exists 
so that 
.
other element of 
equivalence classes to obtain ![$I\in \binom{[24]}{12}$](http://latex.artofproblemsolving.com/8/d/9/8d9352a6f51cc9f7aec204f0714e4acc3cf8fec8.png)
.
We can take the difference of the two sums
First we reindex the second sum so that the indices match by 
Note that for any 
,
,
.
.
.
and 
