A test consisted of 4 questions, each with 3 predetermined possible answers. What is the maximum number of people who could have participated in the test, given that for any 3 test-takers, there was a question where all three marked different answers as correct?
Let the maximum number of contestants be denoted by $a_{4}$, and generally, the maximum number of contestants for $n$ questions be $a_{n}$. It is obvious that $a_{1}=3$. We will show that $a_{n+1} \leqq \frac{3}{2} a_{n}$, if $n \geqq 1$. Consider the $(n+1)$-th question, for which there are three possible answers. Let the number of contestants who gave the 1st answer be $A$, the 2nd answer be $B$, and the 3rd answer be $C$. We will prove that $A+B \leqq a_{n}$. If we select three contestants from $A+B$, there is a question for which all three gave different answers. But this question cannot be the $(n+1)$-th one, because none of them gave the 3rd answer. Therefore, for the $A+B$ contestants, there is always such a question among the first $n$. Hence, $A+B \leqq a_{n}$. Similarly, it can be shown that $B+C \leqq a_{n}$ and $A+C \leqq a_{n}$. Adding these, we get $2(A+B+C) \leqq 3 a_{n}$, which means

$$
a_{n+1}=A+B+C \leqq \frac{3}{2} a_{n}
$$

Since $a_{1}=3$, we have $a_{2} \leqq 1.5 a_{1}=4.5$, and since $a_{2}$ is an integer, $a_{2} \leqq 4$. Then, $a_{3} \leqq 1.5 a_{2}=6$, and thus $a_{4} \leqq 1.5 \cdot 6=9$.

Finally, we will provide an example to show that $a_{4}=9$, that is, 9 contestants can answer the four questions in such a way that they satisfy the problem's conditions. Denoting the answers by $a, b, c$, the following table gives this:

|  | 1st | 2nd | 3rd | 4th |
| :---: | :---: | :---: | :---: | :---: |
| I. | $a$ | $a$ | $b$ | $c$ |
| II. | $a$ | $b$ | $c$ | $a$ |
| III. | $a$ | $c$ | $a$ | $b$ |
| IV. | $b$ | $a$ | $c$ | $b$ |
| V. | $b$ | $b$ | $a$ | $c$ |
| VI. | $b$ | $c$ | $b$ | $a$ |
| VII. | $c$ | $a$ | $a$ | $a$ |
| VIII. | $c$ | $b$ | $b$ | $b$ |
| IX. | $c$ | $c$ | $c$ | $c$ |

Péter Benczúr (Bp., Fazekas M. Gymnasium, 4th grade)