Putnam 2003 A1

Let n be a fixed positive integer. How many ways are there to write n as a sum of positive integers, n = a_1 + a_2 + \cdots + a_k, with k an arbitrary positive integer and a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1? For example with n = 4, there are 4 ways: 4, 2 + 2, 1 + 1+ 2, 1+1+1+1.

Solution: Denote K_n to be the set of tuples of (a_1, \ldots, a_k) with the above properties. We claim that |K_n| = n. We will use induction. It is easy to verify the claim for |K_n| = 1,2,3,4 for n = 1,2,3,4 respectively.

Assume that |K_{l}| = l for some positive integer l, then for a given tuple (a_1, \ldots, a_k) \in K_l we can add 1 to one of the elements in the tuple, and still preserve the property that a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1. If a_1 \not = a_k, then we simply can add 1 where the integers jump, otherwise a_1 = a_2 = \cdots = a_k and we can just have a_k + 1. This gives rise to tuples which are in K_{l+1}. Finally, we have a tuple of 1s to add; this results in |K_{l+1}| = l+1.

We are not done here, as we would need to show that there exists no other tuples in K_{l+1} that we cannot construct as above. This is easy to see, as we can do the inverse operation of subtracting one (with the exception of the tuple of all 1s).

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