Surviving elimination

In every round of a certain game show, $v$ votes are cast by the public to decide which contestants out of $c$ contestants continue to the next round. The contestant with the lowest amount of votes in every round is eliminated. The next round proceeds with $c - 1$ contestants, and so on. What is the minimum number of votes needed to guarantee that a contestant will proceed to the next round, assuming that he/she does not forfeit?

• $c$ is updated at the start of every round to represent the number of remaining contestants.

• $v$ may vary with each round.

• Every round, one contestant must be eliminated by voting, forfeit, or tiebreaker.

• $c \geq 2.$