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Classical blackboard game: Find the invariant!
Tintarn   1
N Nov 16, 2024 by bin_sherlo
Source: Baltic Way 2024, Problem 3
Positive real numbers $a_1, a_2, \ldots, a_{2024}$ are written on the blackboard. A move consists of choosing two numbers $x$ and $y$ on the blackboard, erasing them and writing the number $\frac{x^2+6xy+y^2}{x+y}$ on the blackboard. After $2023$ moves, only one number $c$ will remain on the blackboard. Prove that
\[
c<2024 (a_1+a_2+\ldots+a_{2024}).\]
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Tintarn
Nov 16, 2024
bin_sherlo
Nov 16, 2024
Classical blackboard game: Find the invariant!
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Source: Baltic Way 2024, Problem 3
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Tintarn
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Positive real numbers $a_1, a_2, \ldots, a_{2024}$ are written on the blackboard. A move consists of choosing two numbers $x$ and $y$ on the blackboard, erasing them and writing the number $\frac{x^2+6xy+y^2}{x+y}$ on the blackboard. After $2023$ moves, only one number $c$ will remain on the blackboard. Prove that
\[
c<2024 (a_1+a_2+\ldots+a_{2024}).\]
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bin_sherlo
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