2014 AMC 12B Problems/Problem 13
Problem
Real numbers and
are chosen with
such that no triangles with positive area has side lengths
,
, and
or
,
, and
. What is the smallest possible value of
?
$\textbf{(A)}\ \frac{3+\sqrt{3}}{2}\qquad\textbf{(B)}\ \frac{5}{2}\qquad\textbf{(C)}\ \frac{3+\sqrt{5}}{2}\qquad\textbf{(D)}}\ \frac{3+\sqrt{6}}{2}\qquad\textbf{(E)}\ 3$ (Error compiling LaTeX. Unknown error_msg)
Solution
Notice that . Using the triangle inequality, we find
In order for us the find the lowest possible value for
, we try to create two degenerate triangles where the sum of the smallest two sides equals the largest side.
Thus we get
and
Substituting, we get
Solving for
using the quadratic equation, we get
(Solution by kevin38017)