= Solution
A <chiral superfield> obeys
$$
\boxed{\bar D_{\dot\alpha}\Phi=0}.
$$
Examples in the <Minimal supersymmetric Standard Model> include the quark, lepton and Higgs chiral superfields. Such a multiplet contains a complex scalar $\phi$, a two-component <Weyl spinor> $\psi$, and a complex <auxiliary field> $F$; only the scalar and fermion propagate on shell.
The chiral coordinate $y^\mu=x^\mu+i\theta\sigma^\mu\bar\theta$ is annihilated in the required combination by $\bar D_{\dot\alpha}$, so the solution has the simple form
$$
\boxed{\Phi(y,\theta)=\phi(y)+\sqrt2\theta\psi(y)+(\theta\theta)F(y)}.
$$
Taylor-expand each component about $x$. Nilpotence truncates the series, and the Grassmann identity
$$
(\theta\sigma^\mu\bar\theta)(\theta\sigma^\nu\bar\theta)
=-\frac12(\theta\theta)(\bar\theta\bar\theta)\eta^{\mu\nu}
$$
gives the <chiral-superfield component expansion>
$$
\Phi=\phi+\sqrt2\theta\psi+(\theta\theta)F
+i\theta\sigma^\mu\bar\theta\,\partial_\mu\phi
-\frac{i}{\sqrt2}(\theta\theta)\partial_\mu\psi\sigma^\mu\bar\theta
+\frac14(\theta\theta)(\bar\theta\bar\theta)\Box\phi.
$$
Therefore
$$
\boxed{A=i,\qquad B=-\frac{i}{\sqrt2},\qquad C=\frac14}.
$$
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