Is {a^n (a+b)^n | n>0} a Deterministic CFL?
You needn't determine the end of "first part".

Note $L$ is exactly the set of strings satisfying the following three constraints:


Its length is even.
It only contains $a$ and $b$.
The first $b$ appears in its latter half.


Constraints 1 and 2 are easy to check. To check constraint 3, the DPDA can push a symbol to its stack each time it reads a character until the first $b$ appears (excluding), and then pop a symbol each time it reads a character. Constraint 3 is satisfied if and only if the initial stack symbol is never read during the popping process.