= Solution
Let $\ell_j(\tau)=\prod_{k\ne j}(\tau-c_k)/(c_j-c_k)$ be the <Lagrange interpolation> basis. Set $Y_i=P(t_n+c_i h)$ and $F_j=f(t_n+c_jh,Y_j)$. Since $P'$ is a <polynomial> of degree at most $s-1$, the collocation conditions give
$$
P'(t_n+\tau h)=\sum_{j=1}^s\ell_j(\tau)F_j.
$$
Integrate from zero to $c_i$ and to one to obtain
$$
Y_i=y_n+h\sum_j a_{ij}F_j,\qquad
y_{n+1}=y_n+h\sum_jb_jF_j,
$$
$$
\boxed{a_{ij}=\int_0^{c_i}\ell_j(\tau)d\tau,\qquad
b_j=\int_0^1\ell_j(\tau)d\tau.}
$$
These are the stage and update equations of a <Runge-Kutta method> with nodes $c_i$. Conversely, stages satisfying these equations reconstruct $P$ by integrating the displayed interpolation <polynomial>, so the equivalence works in both directions. Because $\sum_j\ell_j=1$, they also satisfy the <collocation tableau row-sum identity> $\sum_j a_{ij}=c_i$.
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