2023 AMC 10A Problems/Problem 4
Problem
A quadrilateral has all integer sides lengths, a perimeter of 
, and one side of length 
. What is the greatest possible length of one side of this quadrilateral?
![\[\textbf{(A)}~9\qquad\textbf{(B)}~10\qquad\textbf{(C)}~11\qquad\textbf{(D)}~12\qquad\textbf{(E)}~13\]](http://latex.artofproblemsolving.com/c/a/a/caa0c870b300a94184a7cea172ecb5dba7a38f2c.png)
Solution 1
Lets use the triangle inequality. We know that for a triangle, the 2 shorter sides must always be longer than the longest side. Similarly for a convex quadrilateral the shortest 3 sides must always be longer than the longest side. Thus the answer is 
