Let $T$ be the theory of linear, dense order, without minimum or maximum in the language $\mathscr{L}$ . Expend the language by adding it countable amount of constants: $\mathscr{L}^{*}=\mathscr{L}\cup\{c_{0},c_{1},c_{2},...\}$ and let $T^{*}=T\cup\{c_{i}<c_{j}\mid i<j\}$ .
Prove that $T^{*}$ is complete.
I already know that $T$ is complete since $T$ is $\aleph_{0}$ -categorical. Two countable models of $T$ are isomorphic.
On
The Hint in tomasz's answer is a good approach. Here's another one.
By Löwenheim-Skolem, it suffices to show that if $M$ and $N$ are countable models of $T^*$, then $M\equiv N$. Let $\varphi$ be an $L^*$-sentence. Then $\varphi$ is an $L'$-sentence, where $L'$ is a finite sublanguage of $L^*$ (in particular, it only has finitely many of the new constant symbols). Show that $M|_{L'}\cong N|_{L'}$.
Hint: $T$ has quantifier elimination. Since we don't add any non-constant symbols, $T^*$ has it as well.