KöMaL 2020 B5098

Two players, First and Second, are playing the following game: First selects a positive integer not greater than 2020, which Second is trying to find out by guessing (by naming a number as a guess). The possible answers of First are as follows: My number is smaller than that.''; You are right.''; My number is greater than that.'' If the answer is My number is smaller than that'' or You are right'', then Second pays 10 forints (Hungarian currency) to First. If the answer is My number is greater than that'' then he pays 20 forints. What is the minimum possible amount of money that Second needs to have in order to be certain that he can find out the number, and what strategy should he use? (The game terminates with the first ``You are right'' answer, even if Second already knows the number before asking the last question.)