= Solution
Use the standard convention that the <compact> <manifold> is nonempty and has no boundary. Orient it by $\omega^m$. Since $d\omega=0$, every power $\omega^k$ is closed. If $\omega^k=d\eta$ for some $1\le k\le m$, then
$$
\omega^m=d(\eta\wedge\omega^{m-k}),
$$
again by Leibniz and closedness. <Stokes theorem> would give $\int_M\omega^m=0$. But this integral is strictly positive in the symplectic <orientation>, a contradiction. Therefore
$$
\boxed{[\omega^k]\ne0\in H^{2k}_{\mathrm{dR}}(M)\quad(1\le k\le m)}.
$$
The boundaryless hypothesis matters: a <compact> disk has a symplectic area form but no nonzero ordinary second cohomology class.
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