1A. Anastas, Elena, Petar, and Martina, while waiting for the results of the regional mathematics competition, commented on their possible placement in the national competition. Anastas said, "If I place in the national competition, then Elena will also place in the competition." Elena said, "If I place in the national competition, then Petar will also place in the competition." Petar said, "If I place in the national competition, then Martina will also place in the competition." All these statements are true, however, only two students placed in the national competition. Which two students placed from the regional to the national competition?
Solution. Let Anastas have placed in the state competition. Then, since all statements are true, from Anastas's statement - "If I place in the state competition, then Elena will also place," it follows that Elena must have placed in the state competition. From Elena's statement - "If I place in the state competition, then Petar will also place in the competition," it follows that Petar must have placed in the competition. From Petar's statement, "If I place in the state competition, then Martina will also place," it follows that Martina placed in the state competition. In this case, we find that all 4 students placed in the state competition, which is a contradiction. Therefore, Anastas did not place in the state competition.

Let Anastas not have placed, but let Elena have placed in the state competition. From her statement - "If I place in the state competition, then Petar will also place in the competition," it follows that Petar must have placed in the competition. From Petar's statement, "If I place in the state competition, then Martina will also place," it follows that Martina placed in the state competition. In this case, we find that 3 students placed in the state competition, which is a contradiction. Therefore, Elena did not place in the state competition.

Let Anastas and Elena not have placed, but let Petar have placed in the state competition. Then from his statement - "If I place in the state competition, then Martina will also place," it follows that Martina placed in the state competition, so in this case, we have only 2 participants placed, Petar and Martina, and the truth of the first two statements is correct because $\tau(F \Rightarrow p)=T$, where $F$ is a false statement, $T$ is a true statement, and $p$ is any statement. Therefore, only Petar and Martina placed in the state competition.