Question 72
The sport competition consists of 25 contests; in each contest there is exactly one winner who receives a gold medal. 25 athletes participate in this competition, each of them participates in all 25 contests. There are 25 sports experts. Each of 25 experts is to make his *prediction* how many gold medals each athlete will receive; and in his prediction the numbers of medals must be non-negative integers whose sum equals 25. An expert is considered *competent* if he correctly guesses the number of gold medals of at least one athlete. Find the greatest $k$ such that the experts can make their predictions so that at least $k$ of them will be considered competent, regardless of the results of the competition.