what's subtle difference between these three expression?
I am sympathetic to your problems.I am sympathetic about your problems.I am sympathetic with your problems. They all seem grammatically correct to me. But are they have the same meaning?

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Answer

to be + sympathetic to
to have + sympathy with

I am sympathetic to your plight (situation).
I have sympathy with your situation.

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being sympathetic with your problems is the one that sounds best to me. I could say to also, but I think it may get less use. About seems a bit off, but it’s probably ok too.

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The first two sentences are used more in daily life but they all mean the same thing. Here is a link that could help you:

https://lingohelp.me/preposition-after-adjective/sympathetic-to-towards-with-about-for/

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US Englisch may be more flexible with this than the answer Drew gave but I do like his examples as a good guidelines to follow.