Correct solutions were received from Trevor Butenhoff, Jacky Cimarolli,
Chris Coen, Lena Folwaczny, Megan Goers, Nick Hausam, Kim Heintz, Jeremy Light,
Kristina Radlein, Tim Roberts, Scott Small, Todd Stoller. Partial
solutions came from Pat Shea, Adam Brown, John Plachta, Utsav Agarwal. Further correct solutions were
received from Ahron Teitelman, Bill Webb, Alan O'Donnell, Burkart Venzke, Nancy
Schwarzkopf, Ron Welch, Paul Botham, Juan Carlos Marivela. Lou
Cairoli not only gave a correct solution but provided a nice generalization of the
problem.
The maximum sum arises when the numerators are listed in
decreasing order.
Suppose you have a rearrangement containing the summands a/n
and b/(n+1) with a < b.
Simply note that a/n + b/(n+1) < a/(n+1) + b/n
which means that the sum can be increased by transposing the numerators of two adjacent
terms.
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