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Harmonic numbers

High school · ●○○○○ 10 pts

Call three numbers a,b,ca, b, c harmonic when their reciprocals form an arithmetic sequence, that is, 1a−1b=1b−1c\frac{1}{a}-\frac{1}{b}=\frac{1}{b}-\frac{1}{c}. For example, 15,12,1015, 12, 10 are harmonic because 112−115=110−112\frac{1}{12}-\frac{1}{15}=\frac{1}{10}-\frac{1}{12}. Given that xx, 55, 33 are harmonic numbers in this order with x>5x>5, find xx.

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