1951
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1954-1955. Depending on when you were born, early in the year or later.
-- A person who turns 59 on his birthday in 2012 was born in 1953. -- A person who was already 59 when 2012 began was born in 1952.
59 years old
If your brother was 1 year old when you were 2 years old, then he is 1 year younger than you. That makes him 59 when you are 60.
Since the person was born on December 1, 1953, and it is now December 2012, we can subtract the years to get 59 (2012 - 1953 = 59). We know they've had their birthday for that year since December 1 has already passed in December 2012, confirming they are 59 years old.