**C. 964.** In last year's football Champions' League, it happened for the first time ever that four teams of the same country, England (Arsenal, Chelsea, Liverpool and Manchester United), qualified for the best eight. The eight teams were paired up at random, and the winner from each pair qualified for the best four.

*a*) Some English football supporters wished that the English teams would not be paired together, to make it possible for all four of them to qualify for the semi-finals. What was the probability of such a pairing?

*b*) Other English supporters wished that the four English teams would form two pairs, to make sure that two of them would qualify for the semi-finals. What was the probability of such a pairing?

*c*) In reality, two English teams were paired together and the other two teams got non-English opponents. What was the probability of such a pairing?

Suggested by *L. Koncz*

(5 points)

**K. 182.** In a certain town, each inhabitant is either truthful or a liar, and each of them is either sane or insane. Truthful people say what they think and liars say the opposite of what they think. Sane people think what is true, and insane people think the opposite of the truth. Four inhabitants of the town said the following: *Andy:* I am insane. *Mandy:* I am truthful. *Sandy:* I am a liar. *Dandy:* I am sane. *Andy:* Sandy is truthful. *Mandy:* Dandy is insane. *Sandy:* Mandy is a liar. *Dendi:* Sandy is sane. Determine who is truthful and who is a liar, who is sane and who is insane.

(6 points)

This problem is for grade 9 students only.

**K. 185.** Ten teams took part in a football championship. Every team played every other team exactly once. In each game, the winner team gets 3 points and the losers get 0. In the case of a tie, each team gets 1 point. At the end of the championship, the total of the scores of the teams was 119. Is it true that there was at least one team that tied at least four times?

(6 points)

This problem is for grade 9 students only.