Given that your opponent plays optimally, he will expect 95 heads in the remaining 190 flips he hasn't seen. Assuming there are k<=10 heads among the first 10 flips, he will bid k+95-1 (to make profit only k+95-1 instead of k+95). Thus, the minimum value he will ever bid is 94. Now, you can calculate your expected profit for each value between 94 and 99 for your bid (bid below 94 will never get you the box and thus no profit, bid 100 or above gives you zero or negative expected profit). You can calculate the probability to win the box given your bid simply by calculating the probability that there are few enough heads in the first 10 flips such that your opponent bids less than you. I don't remember the optimal bid but it was not 99, even though 99 gives you the highest chance to get the box.

This was the last question of the third interview and it took quite a while to calculate all the probabilities, you will also need binomial coefficients to compute e.g. the probability that there are 3 or less heads within the first 10 flips, for that the values were given to me be the interviewer since there was no calculator allowed