= Solution
For every $m\geq1$, every choice $x_1,\ldots,x_m\in\mathcal X$, and every coefficient <vector> $\alpha\in\mathbb R^m$, the required condition is
$$
\boxed{\sum_{i,j=1}^m\alpha_i\alpha_j k(x_i,x_j)\geq0}.
$$
Thus $\alpha^TK\alpha\geq0$ for the <kernel matrix> $K$. For a <feature map>, this follows from
$$
\sum_{i,j}\alpha_i\alpha_j\langle\varphi(x_i),\varphi(x_j)\rangle=\left\|\sum_i\alpha_i\varphi(x_i)\right\|_{\mathcal H}^2\geq0.
$$
In <kernel methods>, the term <positive-definite kernel> commonly uses this non-strict inequality. Strict positive definiteness is the stronger requirement that the expression be positive for distinct inputs and every nonzero $\alpha$; it is not required here.
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