HMMT November 2009 E2

Given that $a+b+c=5$ and that $1 \leq a, b, c \leq 2$ , what is the minimum possible value of $\frac{1}{a+b}+\frac{1}{b+c}$ ?

Dataset text NuminaMath olympiads. Quoted with source.

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\frac{4}{7}

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