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Many families in Formosa are faced with such a problem.

Many ... face such a problem.

Many ... meet such a problem.

Many ... meet with such a problem.

Many ... are met with such a problem.

Do all of the above wordings sound right? If yes, do they send the same message to you? If not, what are the slightly varied value? Thanks.
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To face a problem or be faced with a problem is a fancier way of saying to have a problem, so those sound right.

Meeting (or meeting with) problems is not English! You meet people, not things -- at least in American English.

CJ
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CalifJimTo face a problem or be faced with a problem is a fancier way of saying to have a problem, so those sound right.

Meeting (or meeting with) problems is not English! You meet people, not things -- at least in American English.

CJ

Thanks, Jim, for your helpful reply.

For what it's worth, is there any subtle difference between to face and to be faced with a problem?
Any difference? Not really. Facing is a symmetric relation. If I face you, you face me, and we face each other, so I am faced by you and you are faced by me, so to speak.

To face a problem, however, suggests something more active, perhaps readiness to solve it.
To be faced with a problem can suggest that the problem has just somehow appeared.

CJ
CalifJimAny difference? Not really. Facing is a symmetric relation. If I face you, you face me, and we face each other, so I am faced by you and you are faced by me, so to speak.

To face a problem, however, suggests something more active, perhaps readiness to solve it.
To be faced with a problem can suggest that the problem has just somehow appeared.

CJ

Thanks, Jim, for the clear answer.

Got it.
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Actually, there is a difference:

To be FACED WITH a problem means that a problem has developed. You now have a problem.

To FACE a problem means that instead of running away or avoiding the problem, you will deal with it and try to solve it.