Every sheet has an odd number on the front and the next even number on the back.  

Let every front page have odd number  i=1,...,43
and every back page have the next even number , i=1,...,43
Thus the 43 sums of the numbers on the 43 pages, front and
back, are , i=1,...43

We are investigating whether the sum of those 43 sums can be 2011.

Suppose 



 

 



Divide both sides by 2



Thus we have reached a contradiction since we have
an even number equalling to an odd number.

So we have shown that the sum cannot be 2011.

Edwin