= Solution
In $\mathbb R^{2n}$, take the two <unit vectors>
$$
v_{\mathrm{even}}=\frac1{\sqrt n}(0,1,0,1,\ldots,0,1),\qquad
v_{\mathrm{odd}}=\frac1{\sqrt n}(1,0,1,0,\ldots,1,0).
$$
Their supports are disjoint, so they are <orthonormal>. Apply part (a) to $a=(a_1,\ldots,a_{2n})$:
$$
\sum_{i=1}^{2n}a_i^2\geq\frac1n\left(\sum_{i=1}^n a_{2i}\right)^2+\frac1n\left(\sum_{i=1}^n a_{2i-1}\right)^2.
$$
Multiplying by $n$ gives
$$
\boxed{n\sum_{i=1}^{2n}a_i^2\geq\left(\sum_{i=1}^n a_{2i}\right)^2+\left(\sum_{i=1}^n a_{2i-1}\right)^2.}
$$
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