Solution:

The answer is 64. We denote by $p_{1}, p_{2}, p_{3}$ the scores that a competitor can obtain on problems $1, 2, 3$, respectively. First of all we observe that certainly the number of competitors cannot exceed 64. Indeed, considering just the first two problems, the possible pairs of scores $\left(p_{1}, p_{2}\right)$ relative to these two problems are $8^{2} = 64$, and therefore, if we want different competitors to correspond to different pairs, the number of competitors cannot exceed 64.

Secondly, however, with 64 competitors the required result can be achieved. Indeed, imagine that the 64 competitors obtain, on the first two problems, all 64 available pairs $\left(p_{1}, p_{2}\right)$ of scores. It is certainly possible, since the scores range between 0 and 7, that all competitors obtain a total score (that is, the sum of the scores $p_{1} + p_{2} + p_{3}$) divisible by 8. In this case there are no two competitors who have the same score on two different problems. Indeed, if two competitors have, for example, equal scores on problem 1 and on problem 3, then, by difference, they must also have an equal score on problem 2, contradicting the assumption made. A completely analogous argument applies to the case in which two competitors have equal scores on problem 2 and on problem 3.

Solution:

The first part of this solution coincides with that of the previous solution. For the second part, we show that 64 competitors can satisfy the required condition. We already know that, on the first two problems, our 64 competitors must obtain all 64 possible combinations of scores. We can therefore construct an $8 \times 8$ table whose rows correspond to the scores of the first problem, and whose columns correspond to those of the second. Each cell of the table will correspond to one of the competitors, and in that cell we want to write the score he obtains in the third problem. The requirement of the exercise then becomes that no row and no column contain two equal numbers. The following table, for example, satisfies this requirement.