= Solution
Write $S=S_0+S_I$, where $S_0$ contains the quadratic <kinetic term> and <mass term>, and $S_I=-\lambda\int d^Dx\,\phi(x)^3/3!$. Then
$$
e^{iS}=e^{iS_0}\sum_{r=0}^\infty\frac{(-i\lambda)^r}{r!(3!)^r}
\int\prod_{a=1}^r d^Dx_a\,\prod_{a=1}^r\phi(x_a)^3.
$$
Let $Z_0=\int D\phi\,e^{iS_0}$ and let $\langle\cdot\rangle_0$ denote the unit-normalized free <Gaussian> <vacuum expectation value>. The unnormalized numerator $\mathcal N_N$ is consequently
$$
\frac{\mathcal N_N}{Z_0}
=\sum_{r=0}^\infty\frac{(-i\lambda)^r}{r!(3!)^r}
\int\prod_{a=1}^r d^Dx_a\,
\left\langle T\phi(y_1)\cdots\phi(y_N)\prod_{a=1}^r\phi(x_a)^3\right\rangle_0.
$$
The Minkowski <path integral> is understood with the usual <scalar Feynman propagator pole prescription>, as a formal <perturbation series>. A real cubic <potential energy> is unbounded below; no nonperturbative stable <vacuum> is being assumed here.
Set $\Delta(x-z)=\langle T\phi(x)\phi(z)\rangle_0$, $K_i(x)=\Delta(y_i-x)$, and $t=\Delta(0)$, with a common <regularization> understood for coincident points. The zero-order one-point term has one <field> and vanishes. At first order, <Wick's theorem> gives three choices for the <contraction> with the external <field>, leaving a <tadpole diagram>:
$$
\langle T\phi(y_1)\phi(x)^3\rangle_0=3K_1(x)t.
$$
The second-order term contains seven <fields>, so it vanishes by the <vanishing odd correlators of a free Gaussian field>. Hence
$$
\boxed{\mathcal N_1=Z_0\left[\frac{-i\lambda}{2}\int d^Dx\,K_1(x)t+O(\lambda^3)\right].}
$$
The coefficient is $3/3!=1/2$, or <symmetry factor> $S=2$ when each diagram is weighted by $1/S$. No <spacetime> integral has been evaluated. The one-point <tadpole diagram> is shown in the first panel of the following figure; its other panels organize <3/c/solution>.
\Image[/past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2016-cubic-correlator-expansion.png]
{title=Cubic one- and two-point Wick contractions; vacuum factors cancel, but the product of two one-point tadpoles remains in the full two-point correlator}
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