Question 556147
.
ABCD is a quadrilateral with AB=8, BC=13, and DC=15. What is the maximum possible integral value of AD?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



        The logic and the answer in the post by tutor @Theo are incorrect.

        A came to bring a correct solution.



Obviously, the length of AD is always less than the sum AB + BC + CD = 8 + 13 + 15 = 36.


This is true due to the axiom of Geometry, which states that a straight line segment 
connecting two points on a plane is always shorter than any polyline connecting these points.


In principle, the length of AD can be as close as desired to the sum of the three sides, 
but never reaches the value of the sum (until the quadrilateral is not degenerated).


So, the length of AD has no maximum in the strict mathematical sense, although it is bounded 
from the top by the value of the sum, 36.


But since the problem asks about the maximum INTEGRAL value of AD, 
this maximum integral length of AD do exist, and is equal to 35 units.