Five positive integers have a mean of 18 and a median of 20, and 22 is their only mode. What is the smallest possible value of the least of these integers?

Sorted: a ≤ $b$ ≤ 20 ≤ $d$ ≤ $e$. A unique mode of 22 forces $d=e=22$ (22 cannot sit below the median). Then $a+b=90-20-44=26$. No other value may repeat, so $b$ ≠ 20, $b$ ≠ a, giving $b$ ≤ 19 and $a=26-b$ ≥ 7. With $b=19,a=7$: the set 7, 19, 20, 22, 22 works.