First, I tried to work out if it was possible to not get two numbers out of three adding up to an even number by using algebra.
To get an even number, you must add together two even numbers or two odd numbers.
E+E=E and O+O=E while O+E=O
This means I could not have two even numbers or two odd numbers as part of the three numbers chosen, but this is impossible since the only possibilities for choosing numbers are:
O,O,O ; E,E,E ; O,O,E ; E,E,O.
I used the same principle for this problem and found out what you need to get three numbers adding up to a multiple of three. I let x be any multiple of three so the only other numbers are x-1 and x+1. The combinations needed to get a multiple of three are:
x+x+x=3x
(x+1)+(x+1)+(x+1)=3x+3
(x-1)+(x-1)+(x-1)=3x-3
(x-1)+x+(x+1)=3x
These results show you cannot have three of any type of number that are represented by x-1, x, x+1. To not use three of a kind, you must use a maximum of two but you cannot get five numbers without using at least one of each, which is also not allowed. This shows it is impossible to get five numbers without having three of them add up to a multiple of three.