This Challenge is derived from GJam 2017 Kickstart Parentheses. This is the first 100 large cases with 1<=L+R<=10^5.
Google Code Jam 2017 Qualifier is March 7, 2017. Typically four challenges with small and large aspects with 27 hours to complete.
The GJam story is given L left and R right parentheses determine the maximum number of parentheses pairings.
Input: [L,R], the quantity of parentheses types
Output: [V], the maximum number of parentheses pairings
Examples: [L,R] [V]; [3,2] 
For the case ()()(, for example, the three balanced groups are () from indexes 1 to 2, () from indexes 3 to 4, and ()() from indexes 1 to 4.
Theory: Brute force permutations and counting will not succeed in a timely manner for L+R=10^5. Determining the inherent mathematical pattern is usually the best way to succeed in GJam. GJam Kickstart solutions(C++,Python).