It depends on what we mean by fair. My first sense is that if you step in and attempt to derive a profit on the back of this being the 10th round of the game, that's unfair. So you would have to chip in money that has no expectation of profit.

-x + .25(330+x) = 0
x = 110

But also, for fairness, your bet shouldn't increase the expectation of anyone already playing.

The normal expectation for that player at this time would be:

-10 + (1/3)(330) = 100, which is cool because everyone is breakeven expectation-wise for having paid up for 10 rounds. If you enter into the race with you 110 entry fee:

-10 + (1/4)(440) = 100, so everything checks out.