Question

Sides of a quadrilateral are all positive integers . Three of them are 5,10,20 . How many possible values are there for the fourth side?

Answer 1

5+10+20 = 35

in order to be a quadrilateral, the 4th side cannot be 35, otherwise you would have a straight line

it also cannot be 0 otherwise you will have a triangle

so it can only be integers from 1 to 34

Answer 2

The length of the fourth side can range from 20-10-5 = 5 to 20+10+5 = 35. These extreme values will give a "quadrilateral" that looks like a line segment. This range includes 31 possible values.

If you insist it have non-zero area, the allowed values range from 6 to 34, a total of 29 possible values.