## Task 3

Ana, Dora, Branko, Cvitko, and Fran took the first five places in the spring half-marathon. When asked which runner took which place, five spectators gave the following answers:

1) Cvitko finished second, and Branko third.
2) Fran finished third, and Dora fifth.
3) Ana finished fourth, and Cvitko second.
4) Dora finished first, and Fran second.
5) Branko finished first, and Ana fourth.

In each answer, one part is correct, and the other is incorrect. Determine the ranking of the top five runners in the spring half-marathon.
## First Solution:

Let's consider all possibilities for Ana:

- If Ana arrived first, then according to the 5th statement, Branko would also be first, which is impossible since they cannot both be first.
- If Ana arrived second, then from the 1st statement, we conclude that Branko is third. From the 2nd statement, Dora must be fifth because Fran cannot be third if Branko is third. According to the 4th statement, Fran is second. Therefore, in the 5th statement, neither part can be true, so Ana cannot be second.
- If Ana arrived third, then from the 1st statement, Cvitko is second. According to the 2nd statement, Dora is fifth. From the 3rd statement, Cvitko is second, which aligns with the previous conclusion. However, in the 4th statement, neither part can be true, which means it is impossible for Ana to be third.
- If Ana arrived fourth, then according to the 3rd statement, Cvitko is not second. From the 1st statement, we conclude that Branko is third. From the 2nd statement, Dora is fifth. According to the 4th statement, Fran is second. Therefore, Cvitko is first. The 5th statement is consistent with all conclusions.
- Let's also consider if it is possible for Ana to be fifth. Then, according to the 2nd statement, Fran is third. From the 3rd statement, Cvitko is second. According to the 4th statement, Dora is first. In the 5th statement, neither part can be true, which means it is impossible for Ana to be fifth.

The only possible solution is that Cvitko is first, Fran is second, Branko is third, Ana is fourth, and Dora is fifth.