Let the numbers be a < b < c < d < e

Then the smallest two, a and b, must have sum 16,

a + b = 16

and the largest two, d and e, must have sum 37,

d + e = 37.

Then the middle number c is equal to 66 - (a+b) - (d+e) = 66 - 16 - 37 = 13.

OK. So, the two smallest numbers a and b give the sum of 16, and the third number is 13.

It means that the next sum, 20 (see the condition) is a+c.

It can not be nothing else.

If a + c = 20 and c = 13, then a = 20 - 13 = 7.

It implies b = 16-7 = 9.

So, the first three numbers are a = 7, b = 9 and c = 13.

OK. Now we can make the similar analysis from the other end.

So, the two largest numbers d and e give the sum of 37, and the third number in the descending order is 13.

It means that the next from the largest sum, 34 is e+c.

It can not be nothing else.

If e + c = 34 and c = 13, then e = 34 - 13 = 21.

It implies d = 37-21 = 16.

Thus the numbers are 7, 9, 13, 16 and 21.

Their product is 275184.

hope this helps!