### Instruction: Given that $a + b + c = 5$ and $1 \le a,$ $b,$ $c \le 2,$ find the minimum value of
\[\frac{1}{a + b} + \frac{1}{b + c}.\].### Response: I notice that the expression $\frac{1}{a + b} + \frac{1}{b + c}$ is symmetric in $a$ and $c,$ so I can assume without loss of generality that $a \le c.$
Then I want to minimize the sum of two fractions, each of which has a denominator that depends on $b.$
I recall that the sum of two fractions is smallest when their denominators are as large as possible, so I want to maximize $a + b$ and $b + c.$
Since $a + b + c = 5,$ I can write $b = 5 - a - c.$
Then the expression becomes $\frac{1}{5 - c} + \frac{1}{5 - a}.$
Now I see that the expression is decreasing in both $a$ and $c,$ so I want to make them as large as possible within the given constraints.
The largest possible value for $a$ and $c$ is $2,$ so I plug that in and get $\frac{1}{5 - 2} + \frac{1}{5 - 2} = \frac{2}{3}.$