## Task 4 - 180614

Three pioneers from a school, Klaus, Silvia, and Frank, were delegated to the district mathematics olympiad for young mathematicians and won a first, a second, and a third prize. When later Rainer was asked about the performance of his classmates, he said:

"I believe Silvia did not win the first prize, Klaus did not get the second prize, which Frank did."

It subsequently turned out that exactly one of Rainer's three statements was true, while the other two were false.

Which of the three named pioneers received the first, which the second, and which the third prize?
Assuming the statement "Frank received a second prize" is true.

Then the other two statements must be false. This would mean that Klaus also received a second prize, which contradicts the problem. Therefore, the considered statement is false.

Assuming the statement "Klaus did not receive a second prize" is true.

Then the other two statements must be false. This would mean that no one received a second prize, which contradicts the problem. Therefore, this statement is also false.

Thus, only the statement "Silvia did not receive a first prize" can be true. Since this makes the other two statements false, Klaus received a second prize.

Furthermore, only Frank could have received a first prize, and thus Silvia must have received a third prize. Only with this prize distribution is exactly one of Rainer's statements true, and the other two are false.

Solutions of the I. Round 1978 adopted from [5]

# 3.19.2 II. Round 1978, Class 6

#### Abstract