= Solution
Define $F_i(x)=a_i+\frac16\sum_{j\ne i}\sin x_j$. Solving the system is equivalent to finding a <fixed point> of $F:\mathbb R^3\to\mathbb R^3$. The <mean value theorem> implies $|\sin u-\sin v|\leq|u-v|$. In the specified metric,
$$
\begin{aligned}
d(F(x),F(y))&\leq\frac16\sum_{i=1}^3\sum_{j\ne i}|x_j-y_j|\\
&=\frac13\sum_{j=1}^3|x_j-y_j|=\frac13d(x,y).
\end{aligned}
$$
Each coordinate difference occurs exactly twice. This metric makes $\mathbb R^3$ a nonempty <complete metric space>, because a <Cauchy sequence> in it is Cauchy in each real coordinate, and coordinate convergence implies convergence in the sum metric. Thus $F$ is a <contraction mapping> with constant $1/3$.
\b[There is exactly one solution for every choice of $(a_1,a_2,a_3)$], by the <contraction mapping theorem>. Iteration of $F$ from any starting vector also constructs it.
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