Recover five numbers from their ten pair sums

Five real numbers $a,b,c,d,e$ (repeats allowed) have the following ten pairwise sums, listed here from largest to smallest: $22,\ 22,\ 19,\ 16,\ 14,\ 13,\ 13,\ 11,\ 11,\ 5.$ What do the squares of the five numbers add up to?

$\tfrac{1405}{4}$. Each number appears in four sums, so the total is $146/4=36.5$. With $a\le b\le c\le d\le e$: $a+b=5$, $d+e=22$ gives $c=9.5$; $a+c=11$ and $c+e=22$ give $a=1.5$, $e=12.5$, hence $b=3.5$, $d=9.5$. These reproduce the list, and the squares add to $351.25=\tfrac{1405}{4}$.