= Solution
Let $A=\{a_1,\ldots,a_n\}\subseteq M$ and let $p(x/A)$ be a type. In an elementary extension choose $c$ realizing it and consider the empty-set type $q=\operatorname{tp}(a_1,\ldots,a_n,c)$. By hypothesis, $M$ contains a tuple $(b_1,\ldots,b_n,d)$ realizing $q$. The map $b_i\mapsto a_i$ is partial elementary, so omega-homogeneity extends it to include $d\mapsto d'$. Then $d'$ realizes $p$. Hence $M$ is omega-saturated, as stated by <homogeneity plus realization of empty-set types implies saturation>.
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