40th Putnam 1979

Problem A1

Find the set of positive integers with sum 1979 and maximum possible product.

Solution

Easy

For n > 4, 2(n - 2) > n, so the maximum product cannot include any integer greater than 4. Also, 2³ < 3², so it cannot include more than two 2s. Since 4 = 2·2, it cannot include both a 2 and a 4. It obviously does not include a 1, since n + 1 > n x 1. So the maximum product must be made up mainly of 3s, with either no, one or two 2s (or equivalently one 4). Hence for 1979 = 2 + 659·3, the maximum product is 3⁶⁵⁹2.

40th Putnam 1979

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