Problem 6.2. In a running competition, five athletes participated: $A, B, C, D$ and $E$. Two predictions were made about the order in which they would finish.

- First prediction: $A$ - first, $B$ - second, $C$ - third, $D$ - fourth, $E$ - fifth.
- Second prediction: $C$ - first, $E$ - second, $A$ - third, $B$ - fourth, $D$ - fifth. It turned out that in the first prediction, the positions of exactly three athletes were correctly predicted, and in the second prediction, the positions of exactly two athletes were correctly predicted. Who took which place in the race?
Answer: $C$ - first, $B$ - second, $A$ - third, $D$ - fourth, $E$ - fifth.

Solution. In the first forecast, the positions of exactly three athletes were correctly predicted, which means the positions of two were incorrectly predicted. Therefore, the positions of these two athletes need to be swapped.

Additionally, in the second forecast, the position of each of the five athletes is not the same as in the first forecast. Therefore, the positions of the three athletes correctly predicted in the first forecast are incorrectly predicted in the second forecast. Then, since the second forecast correctly predicted the positions of two athletes, they must be precisely the two whose positions were incorrectly predicted in the first forecast.

Let's start by considering which pairs of athletes need to swap their positions in the first forecast. In the second forecast, the positions of these two should match the swapped positions in the first.

If these athletes are $A$ and $B$, then, by swapping their positions, we get that in the second forecast, $A$ should be second, and $B$ should be first. But this is not the case.

If these athletes are $A$ and $C$, then, by swapping their positions, we see that in the second forecast, $A$ is third, and $C$ is second. Then the positions of the other three athletes $B, D, E$ are correctly predicted in the first forecast: $B$ - second, $D$ - fourth, $E$ - fifth. This variant satisfies the conditions of the problem.

If these athletes are $A$ and $D$, then, by swapping their positions, we get that in the second forecast, $A$ should be fourth, and $D$ should be first. But this is not the case.

Similarly, by considering the other possible pairs $A$ and $E, B$ and $C, B$ and $D, B$ and $E, C$ and $D, C$ and $E, D$ and $E$, it is not difficult to verify that $A$ and $C$ are the only possible pair whose positions were incorrectly predicted in the first forecast.

Another solution. Note that for each athlete, the forecasts give different results, meaning that in total, they could have guessed correctly no more than once for each athlete. Since the forecasts guessed correctly five times in total for all five athletes, they guessed correctly exactly once for each athlete. Additionally, this means that they guessed correctly exactly once for each position.

Therefore, if the first forecast had guessed correctly about the first position for $A$, the second forecast would not have guessed correctly about $C$ (since $C$ was predicted to be in the first position); then the first forecast would have guessed correctly about $C$.

Similarly, if the first forecast had guessed correctly about $B$, the first forecast would have guessed correctly about $E$ and, similarly, about $D$.

Reasoning this way, it becomes clear that the first forecast either guessed correctly about both $A$ and $C$ or did not guess correctly about either; similarly for $B, E$, and $D$. Since the first forecast guessed correctly about three athletes, it guessed correctly about $B, D$, and $E$, but not about $A$ and $C$. Conversely, the second forecast guessed correctly about $A$ and $C$, but not about $B, D$, and $E$.