USAMO 2006 — Days 1 and 2 Question 2

Let $k > 0$ be a fixed integer. Compute the minimum integer $N$ (in terms of $k$) for which there exists a set of $2k+1$ distinct positive integers that has sum greater than $N$, but for which every subset of size $k$ has sum at most $N/2$.

Proposed by Dick Gibbs (credited in the supplied source).