 Solution

For a fixed horse $i$, independence of the race results gives probability $p_i^m$ of winning every race. The events that different horses win every race are disjoint, soThis is a homogeneous polynomial of degree $m$. Apply part (a) to the nonnegative numbers $p_i$, whose sum is one, to obtain $\boxed{Q\ge N^{1-m}}$. For $m>1$ the smallest probability occurs when every horse has winning probability $1/N$; for a single race $Q=1$ regardless of these probabilities.