= Solution
The <core of a cooperative game> is the set of efficient allocations for which every <coalition> receives at least its own value: $x(S)\ge v(S)$ for every $S\subseteq N$. The singleton constraints give individual rationality, and the remaining proper-coalition inequalities are
$$
x_1+x_2\ge3,\qquad x_1+x_3\ge10,\qquad x_2+x_3\ge6.
$$
Efficiency turns the second into $x_2\le2$; together with $x_2\ge2$ this forces $x_2=2$. Put $x_1=t$ and $x_3=10-t$. The remaining constraints give $t\ge1$, $t\le6$, and $t\le7$ from $x_3\ge3$. The first pair constraint adds only $t\ge1$. Hence
$$
\boxed{C(v)=\{(t,2,10-t):1\le t\le6\}.}
$$
Each point in this segment satisfies all singleton, pair, and grand-coalition constraints, so the description is sufficient as well as necessary. Its endpoints are $(1,2,9)$ and $(6,2,4)$.
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