7
Part of 2020 Tuymaada Olympiad
Problems(2)
Products of digits + 1 is N+1
Source: 2020 Tuymaada Junior P7
10/6/2020
How many positive integers in the segment are such that if all their digits are increased by and then multiplied, the result is ?(F. Bakharev)
number theoryDigits
Police and thieves under surveillance
Source: Tuymaada 2020 Senior, P7
10/6/2020
Several policemen try to catch a thief who has accomplices. To that end they place the accomplices under surveillance. In the beginning, the policemen shadow nobody. Every morning each policeman places under his surveillance one of the accomplices. Every evening the thief stops trusting one of his accomplices The thief is caught if by the -th evening some policeman shadows exactly those accomplices who are still trusted by the thief. Prove that to guarantee the capture of the thief at least policemen are needed.
combinatoricspigeon holealgorithm