The result is trivial for 4 cities or less, so assume there are at least 5 cities. Denote $f(XY)$ as the price of the flight from $X$ to $Y$. Let $K$ be the capital city. Take 4 arbitrary cities $A,B,C,D$. Let the other cities be $E_1,\ldots,E_n$. Consider two routes $A-B-C-D-E_1-E_2-\ldots-E_n$ and $A-C-B-D-E_1-E_2-\ldots-E_n$. These two routes cost the same, so we get $f(AB)+f(CD)=f(AC)+f(BD)$. Therefore we get $f(KB)+f(AC)=f(KC)+f(AB)$ and $f(KA)+f(CD)=f(AC)+f(KD)$. So $f(KA)+f(KB)-f(AB)=f(KA)+f(KC)-f(AC)=f(KD)+f(KC)-f(CD)$ Thus $f(KX)+f(KY)-f(XY)$ is constant for any two cities $X,Y$, let this constant be $C$. Also let $C'$ be the cost of a round trip landing in every city exactly once. Consider an arbitrary round trip that miss the capital city. If we insert a landing in $K$ after landing in $A$ and before landing in $B$, the cost is increased by $f(AK)+f(BK)-f(AB)=C$, which becomes $C'$. So the cost of the round trip that miss the capital is $C'-C$, which is constant. Our proof is complete.