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SAUDI ARABIAN IMO Booklet 2023

Saudi Arabia 2023 counting and probability

Problem

Each cell of a table contains one of the numbers or . For every row we calculate the sum of its entries and for every column we calculate the product of its entries. Can the eight obtained results be all different?
Solution
But both and can appear only as products, which forces a column of 's and a column of 's. And this leaves only three possible row sums, contradiction.
Final answer
No

Techniques

Pigeonhole principleColoring schemes, extremal arguments