= Solution
The <Runge-Kutta method> has update $y_{n+1}=y_n+h\sum_i b_i k_i$. Expanding its squared <Euclidean norm> gives the exact identity
$$
\lVert y_{n+1}\rVert^2-\lVert y_n\rVert^2=h\left[2\sum_i b_i y_n^Tk_i+h\left\lVert\sum_i b_i k_i\right\rVert^2\right].
$$
The stated per-step relation makes this difference zero. Induction from the initial vector then proves
$$
\boxed{\lVert y_n\rVert=\lVert y_0\rVert\quad\text{for all computed steps}.}
$$
For a positive step size the relation is also necessary for exact per-step norm conservation. This algebraic identity applies whenever the stage equations define a step; it does not silently assume solvability of arbitrary implicit stages.
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