US passenger declaration currency requirement

Legally you need to declare any and all currency or negotiable monetary instruments when arriving in OR DEPARTING FROM the US IF their total value is over US$10,000. This amount is the combined value for all passengers travelling together, NOT per person.

Thus if between you and your mother you are carrying more than US$10,000 in cash/etc (regardless of the form or currency) then you will need to declare it. Normally when departing the US there is no departure form that needs to be completed, however this does not negate the need to declare the currency to the CBP officers at the airport. (Specifically you will need to complete form FinCEN 105.)

If you do not declare the currency, and you are caught, then you may need to forfeit the entire amount, so the risks of not declaring the currency are high.

Note that in general there are no rules against actually carrying more than $10,000 cash - the reporting requirements exist to help track money laundering and illegal actives. As long as you have a valid reason for having/carrying the cash, and you correctly declare it, then there will be no issues, and no fees/taxes/etc to be paid on it.

You will need to do a joint declaration when you land in Korea, for currency you are carrying with you into Korea.

On departure from the US, there are no formal requirements - however, they do random checks with currency sniffing dogs.

Nothing to be alarmed about - you just have to show the source of the funds you carry. A simple bank receipt showing the amount would suffice.