= Solution
Independence makes
$$
X\longrightarrow X+Y\longrightarrow X+Y+Z
$$
a <Markov chain>. The <data processing inequality for mutual information> therefore gives $I(X;X+Y+Z)\leq I(X;X+Y)$. Translation by the known value of $X$ is a <bijection>, so <conditional entropy> and independence give
$$
I(X;X+Y+Z)=H(X+Y+Z)-H(Y+Z),
$$
and $I(X;X+Y)=H(X+Y)-H(Y)$. Rearranging proves the stated <entropy submodularity for three independent sums>:
$$
H(X+Y+Z)+H(Y)\leq H(X+Y)+H(Y+Z).
$$
Solved by gpt-5.6-sol high.
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