10. $N$ teams participate in a hockey league, each team plays a match against every other team exactly once. It is known that among any 3 teams, not every pair of teams has a draw. Find the maximum number of draws.
(53rd Belarusian Mathematical Olympiad (Final C Category))
Let $d_{i}$ be the number of draws for the $i$-th team, and let $k=\max _{1 \leqslant i \leqslant v} d_{1}$. Without loss of generality, assume $d_{v}=k$, and the $N$-th team draws with the $i$-th team $(1 \leqslant i \leqslant k)$.

By the given conditions, the $i$-th team and the $j$-th team $(1 \leqslant i, j \leqslant k)$ cannot draw, so the teams from $k+1, \cdots, N-1$ can each draw at most $k$ games. Therefore, the number of draws does not exceed $k(N-k)$. Hence, the maximum number of draws is
$$
\begin{aligned}
M & =\max _{1 \leqslant k \leqslant N} k(N-k) \\
& =\left\{\begin{array}{ll}
m^{2}, & \text { when } N=2 m, \\
m(m+1), & \text { when } N=2 m+1 .
\end{array}\right.
\end{aligned}
$$

The following example shows that $M$ can be achieved.
Let the 1st, 2nd, ..., $m$-th teams draw with the $m+1$-th, ..., $N$-th teams, the $p$-th team beat the $i$-th team $(1 \leqslant i<p \leqslant m)$, and the $q$-th team beat the $j$-th team $(m+1 \leqslant j<q \leqslant N)$, satisfying the number of draws is $M$ and the conditions are met.