We know the year has to be less than 2001 since we already have the palindrome for 10/02. It can't be any year in 1900 because that would result in a day of 91. Same for 1800 down to 1400. It could be a year in 1300 because that would be the 31st day. So what's the latest year in 1300 that would make a month? You might think it would be 1321, since that would give us the 12th month, but we have to remember that we want the maximum year in the 1300 century with a valid month, which would actually be 1390, since 09/31 is a valid date.
But of course, there's a catch, there are not 31 days in september, only 30. So we have to go back to august 08 which means the correct date would be 08/31/1380.