Problem:
Real numbers a and b are chosen with 1<a<b such that no triangle with positive area has side lengths 1,a, and b or b1​,a1​, and 1 . What is the smallest possible value of b ?
Answer Choices:
A. 23+3​​
B. 25​
C. 23+5​​
D. 23+6​​
E. 3 Solution:
There is a triangle with side lengths 1,a, and b if and only if a>b−1. There is a triangle with side lengths b1​,a1​, and 1 if and only if a1​>1−b1​, that is, a<b−1b​. Therefore there are no such triangles if and only if b−1≥a≥b−1b​. The smallest possible value of b satisfies b−1=b−1b​, or b2−3b+1=0. The solution with b>1 is 21​(3+5​)​. The corresponding value of a is 21​(1+5​).