2002 AIME I Problems/Problem 3
Jane is 25 years old. Dick is older than Jane. In years, where is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let be Dick's present age. How many ordered pairs of positive integers are possible?
Let Jane's age years from now be , and let Dick's age be . If , then . The possible pairs of are:
That makes 36. But , so we subtract all the extraneous pairs: and .
Start by assuming that (essentially, Jane is in the 20s when their ages are 'reverses' of each other). Then we get the pairs Repeating this for the 30s gives From here, it's pretty clear that every decade we go up we get as a pair. Since both ages must always be two-digit numbers, we can show that each decade after the 30s, we get 1 fewer option. Therefore, our answer is
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