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Welcome to FT!

Although you would normally not earn many miles by spending $5-10K, that amount of money will satisfy the spend requirement on some credit cards that offer hefty sign-up bonuses.

Suggest you look at the thread in the FT MilesBuzz! forum on the current Citi/AA credit-card offers. Also consider the current Chase/UA bonus offer.

Since both AA and Chase offer one-way awards, you might very well use UA miles in one direction, and AA miles in the other.

Also take a look at the FT AA forum, for information on AA's gateway stopover option. If you can secure award seats on a nonstop flight from Europe to LAX on AA or any of its partner carriers, you can then "stop over" at LAX, and continue on -- months later -- on a one-way flight to another North American destination (like Hawaii) for no additional mileage. (To get back to California from Hawaii, you would either have to spend cash or more miles, however.)

Good luck!