= 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, so
$$
\boxed{Q=\sum_{i=1}^N p_i^m.}
$$
This is a homogeneous <polynomial> of degree $m$. Apply part (a) to the nonnegative numbers $p_i$, whose sum is one, to obtain \b[$\boxed{Q\geq N^{1-m}}$]. 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>.
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