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335

Part of 1956 Moscow Mathematical Olympiad

Problems(1)

MMO 335 Moscow MO 1956 100 numbers in a row, sum conditions, question

Source:

8/19/2019
a) 100100 numbers (some positive, some negative) are written in a row. All of the following three types of numbers are underlined: 1) every positive number, 2) every number whose sum with the number following it is positive, 3) every number whose sum with the two numbers following it is positive. Can the sum of all underlined numbers be (i) negative? (ii) equal to zero?
b) nn numbers (some positive and some negative) are written in a row. Each positive number and each number whose sum with several of the numbers following it is positive is underlined. Prove that the sum of all underlined numbers is positive.
algebracombinatoricsSum