I should get to solving the math problems.
Does the above sentence mean- I should start solving the math problems? And is it grammatically correct?
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Yes, the sentence is correct, and you got the meaning right.
There is a nuance: “I should get to doing …” implies that you might have started doing it sooner. It's somewhat intermediate between “I should start doing …” which is neutral, and “I should get around to doing …”.
This expresssion is not to be confused with “I should get to do …”, which has a completely different meaning: “I should have the opportunity to do …”.