= Solution
Let $\Phi_t$ be the local <flow of a vector field> $V$. The <Lie derivative of a differential form> is $\mathcal L_V\eta=\left.\frac{d}{dt}\right|_{t=0}\Phi_t^*\eta$. We prove <Cartan's magic formula>
$$
\mathcal L_V=d\iota_V+\iota_Vd.
$$
Both sides are degree-zero <derivations of an algebra> on the <exterior algebra> of <differential forms>. They agree on a <function> $f$, since $\iota_Vdf=V(f)$. They also agree on $df$: pullback commutes with the <exterior derivative>, so $\mathcal L_Vdf=d(Vf)$, while $(d\iota_V+\iota_Vd)df=d(Vf)$. Locally every <differential form> is a sum of products of <functions> and coordinate differentials, proving the identity in every degree.
For a compactly supported <Hamiltonian vector field> $V_H$, take the convention $\iota_{V_H}\omega=-dH$. Since $d\omega=0$, <Cartan's magic formula> gives $\mathcal L_{V_H}\omega=0$, so its complete <flow of a vector field> consists of <symplectomorphisms>. Given $p$ and $v\in T_pM$, a <bump function> lets us choose a compactly supported $H$ with $dH_p=-\iota_v\omega_p$, and hence $V_H(p)=v$. Choose such fields for a <basis> of $T_pM$. Their successive small-time flows give a map with invertible derivative at the origin. The <inverse function theorem> makes the orbit of $p$ open. Every orbit is open; because $M$ is <connected>, there is only one orbit. Thus \b[the compactly supported symplectomorphisms act transitively on points], by <Hamiltonian transitivity on a connected symplectic manifold>.
On the unit <sphere>, the standard <symplectic area> is $4\pi$. The equator has complementary disks of areas $2\pi,2\pi$, whereas the latitude at height $1/2$ has complementary disks of areas $\pi,3\pi$: the cap above height $h$ has area $2\pi(1-h)$. A <symplectomorphism> must preserve the unordered pair of complementary areas. Therefore \b[no such symplectomorphism exists], by <complementary areas obstruct symplectic equivalence of separating curves>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-20-sphere-area-obstruction.png]
{title=Complementary sphere areas distinguish the equator from the latitude at height one half}
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