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Part of 2024 Poland - Second Round

Problems(1)

Rational sequence

Source: Polish MO Second round 2024 P1

Does there exist a rational

x_1

, such that all members of the sequence

x_1, x_2, \ldots, x_{2024}

x_{n+1}=x_n+\sqrt{x_n^2-1}

n=1, 2, \ldots, 2023

are greater than

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