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 E1,…,En. Consider two routes A−B−C−D−E1−E2−…−En and A−C−B−D−E1−E2−…−En. 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.