**B. 4310.** Let be positive numbers, such that *a*_{k+1}-*a*_{k}1 for all *k*=0,1,...,*n*-1. Show that

(IMC 2010 -- Blagoevgrad, Bulgaria)

(5 points)

**B. 4311.** *P* is a given point in the interior of the acute-angled triangle *ABC*. The lines *AP*, *BP* and *CP* intersect the opposite sides at the points *A*_{1}, *B*_{1} and *C*_{1}, respectively. Given that *PA*_{1}=*PB*_{1}=*PC*_{1}=3 and *AP*+*BP*+*CP*=43, prove that *AP*^{.}*BP*^{.}*CP*=441.

(*A. Máder* and *V. Vigh,* Szeged)

(4 points)

**K. 265.** A man commutes to work by train. Every day in the afternoon, he arrives to the railway station with the same train, and his wife drives him home. The wife leaves home just in time to arrive at the station when her husband's train comes in. One day, the man arrived at the station an hour earlier, and started to walk home along the road on which his wife drove him home every day. The wife left home at the usual time. When she saw her husband on the road, she immediately stopped, picked him up and turned around. They arrived home 20 minutes earlier than usual. Calculate the time in minutes that the husband walked along the road.

(6 points)

This problem is for grade 9 students only.

**K. 270.** A fair bookmaker only takes fair bets. For example, if you bet 3 to 5 on something, it means that you pay 3 if you lose and get 5 if you win, but in the long run no one gains anything on this bet. There are three horses running a race: Curse, Worse, and Hearse. The fair bookmaker takes 2 to 1 bets on Hearse winning the race, and 3 to 7 bets on Curse winning. (These are fair bets.) In what ratio can one bet on Worse coming in first?

(6 points)

This problem is for grade 9 students only.