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MMO 139 Moscow MO 1947 like pascal triangle

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July 20, 2019
pascal triangleSumcombinatoricsEvenPascal's Triangle

Problem Statement

In the numerical triangle ................1..............................1.............. ...........1...1...1....................1 ...1 ...1......... ......1...2...3...2...1..........1... 2... 3 ... 2 ... 1.... .1...3...6...7...6...3...1.1...3...6...7...6...3...1 .............................................................. each number is equal to the sum of the three nearest to it numbers from the row above it; if the number is at the beginning or at the end of a row then it is equal to the sum of its two nearest numbers or just to the nearest number above it (the lacking numbers above the given one are assumed to be zeros). Prove that each row, starting with the third one, contains an even number.