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SAUDI ARABIAN IMO Booklet, 2023

Combinatorics Difficulty 7.7 National Olympiad, round 2 Prove it Saudi Arabia

Each cell of a $4 \times 4$ table contains one of the numbers $1$ or $2$ . 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 $1$ and $16$ can appear only as products, which forces a column of $1$ 's and a column of $2$ 's. And this leaves only three possible row sums, contradiction. $\square$

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