A math competition consists of three problems, each of which receives an integer score from 0 to 7. For any two competitors, we know that there is at most one problem on which they obtained the same score. Find the largest possible number of competitors in this competition.
Solution

There are 8 possible scores for each problem, and consequently, $8 \cdot 8=64$ distinct possible scores for the first two problems. Since no two competitors can have exactly the same scores on the first two problems, the total number of competitors cannot be greater than 64. We will now show that this maximum value is achievable. To do this, it suffices to exhibit a distribution of scores among 64 players that satisfies the conditions of the problem. Consider the table:

|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 0 |
| 2 | 2 | 3 | 4 | 5 | 6 | 7 | 0 | 1 |
| 3 | 3 | 4 | 5 | 6 | 7 | 0 | 1 | 2 |
| 4 | 4 | 5 | 6 | 7 | 0 | 1 | 2 | 3 |
| 5 | 5 | 6 | 7 | 0 | 1 | 2 | 3 | 4 |
| 6 | 6 | 7 | 0 | 1 | 2 | 3 | 4 | 5 |
| 7 | 7 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

The rows will indicate the score on the first problem, and the columns the scores on the second problem. For each of the 64 possible combinations of rows and columns, which achieve all 64 possible scores on the first two problems, choose the score for the third problem as the number written at their intersection. For example, the combination of the row numbered 2 with the column numbered 5 generates the score $(2,5,7)$. Since there are no repeated numbers in the rows and columns, all these triples generate scores satisfying the conditions of the problem.

Note: Another possible construction would be to choose for each of the 64 pairs of possible scores on the first two problems, say $\left(p_{1}, p_{2}\right)$, use as $p_{3}$ the unique integer in the set $\{0,1, \ldots, 7\}$ such that $p_{1}+p_{2}+p_{3}$ is a multiple of 8. If there are two triples $\left(p_{1}, p_{2}, p_{3}\right)$ and $\left(q_{1}, q_{2}, q_{3}\right)$ with two scores the same on the same problems, then the other scores must also coincide, as the set $\{0,1, \ldots, 7\}$ contains only one representative of each possible remainder in division by 8.