ASU 221 All Soviet Union MO 1976 rows of 1000 numbers on blacboard
Source:
July 5, 2019
game strategycombinatoricsblackboard
Problem Statement
A row of numbers is written on the blackboard. We write a new row, below the first according to the rule: We write under every number the natural number, indicating how many times the number is encountered in the first line. Then we write down the third line: under every number -- the natural number, indicating how many times the number is encountered in the second line, and so on. a) Prove that there is a line that coincides with the preceding one. b) Prove that the eleventh line coincides with the twelfth. c) Give an example of the initial line such, that the tenth row differs from the eleventh.