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.

• \( c \geq 2. \)