FINAL ROUND

Question

A sequence is called *N-nice* if it consists of a finite number of the consecutive positive integers and the sum of all its terms is equal to N. Find the total number of all 4030^k-nice sequences, where k is a positive integer. (*N*-nice sequence may consist of a single term.)

Accepted Answer

Let P(N) be the number of N-nice sequences, D(N) be the number of odd divisors of N. Then P(N) = D(N) (see the solution of Problem C.7). Since D(4030^k) = D(2^k 5^k 13^k 31^k) = (k+1)^3, we see that the number of the required 4030^k-nice sequences is equal to (k+1)^3.