Let $A$ and $B$ be the respective sums. Then $B$ is at least $1+2+3+\ldots+15 = 120$, so $A$ is at least $3 \cdot 120 = 360$. But $A$ is at most $17+18+\ldots+31 = 360$, so $A$ must be $360$ and $B$ must be $120$. Therefore Alex must pick $17, 18, \ldots, 31$ and Bogdan has to pick $1, 2, \ldots, 15$, so the remaining card must have $16$ written on it.