1

Part of 1999 All-Russian Olympiad

Problems(3)

Sum of digits of 9A

Source: All-Russian MO 1999

The decimal digits of a natural number

A

form an increasing sequence (from left to right). Find the sum of the digits of

9A

Pooh and Piglet cooperate against Rabbit

Source: All-Russian MO 1999

There are three empty jugs on a table. Winnie the Pooh, Rabbit, and Piglet put walnuts in the jugs one by one. They play successively, with the initial determined by a draw. Thereby Winnie the Pooh plays either in the first or second jug, Rabbit in the second or third, and Piglet in the first or third. The player after whose move there are exactly 1999 walnuts loses the games. Show that Winnie the Pooh and Piglet can cooperate so as to make Rabbit lose.

combinatorics unsolvedcombinatoricsGame Theory

19 numbers totalling 1999

Source: All-Russian MO 1999

19

distinct natural numbers with equal sums of digits, whose sum equals

1999

number theory unsolvednumber theory