2023 AIME II Problems/Problem 5
Since 55r is greater than r, we know 55 r can be simplified. Call r . This means 55r is
. Since 55r can clearly be simplified, let's look at the possibilities, where b is divisible by factors of 55.
Case 1: B is divisible by 55.
If this is the case, the sum of the numerator and denominator of 55r is a + . This clearly cannot equal a+b, so this case is invalid.
Case 2: B is divisible by 11.
If this is the case, the sum of the numerator and denominator of 55r is 5a + . This is equal to a+b. Solving this equation gives
= r =
.
Case 3: B is divisible by 5.
In this case, the sum of the numerator and denominator of 55r is 11a + . This is equal to a+b, and solving that equation gives
=
= r.
Clearly, B is divisible by 1 doesn't make sense, so these are the only possibilities. Summing these fractions give , so the answer is 169 + 550 = 719.
~Anonymus