grammar of "try as I might"

Solution 1:

Try as someone might is a set phrase that is used in its 'fixed form' also in the past tense:

• used for saying that someone is making a very great effort but still cannot do something
  • Try as he might, he could not forget.

(MacMillan Dict)

The expression exists also in the less common form of:

Try as I may:

• Cliché a phrase that introduces an expression of regret or failure.
  • Bill: Try as I may, I cannot get this thing put together right. Andy: Did you read the instructions? Rachel: Wow! This place is a mess! Mother: Try as I might, I can't get Andrew to clean up after himself.

(McGraw-Hill Dictionary of American Idioms and Phrasal Verbs)

Ngram : try as I/he/she/they might