Difference between revisions of "2023 AIME II Problems/Problem 5"
(Created page with "Let <math>S</math> be the set of all positive rational numbers <math>r</math> such that when the two numbers <math>r</math> and <math>55r</math> are written as fractions in lo...") |
|||
Line 1: | Line 1: | ||
− | + | Since 55r is greater than r, we know 55 r can be simplified. Call r <math>\frac{a}{b}</math>. This means 55r is <math>\frac{55a}{b}</math>. 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 + <math>\frac{b}{55}</math>. 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 + <math>\frac{b}{11}</math>. This is equal to a+b. Solving this equation gives <math>\frac{a}{b}</math> = r = <math>\frac{5}{22}</math>. | ||
+ | |||
+ | Case 3: B is divisible by 5. | ||
+ | |||
+ | In this case, the sum of the numerator and denominator of 55r is 11a + <math>\frac{b}{5}</math>. This is equal to a+b, and solving that equation gives <math>\frac{a}{b}</math> = <math>\frac{2}{25}</math> = r. | ||
+ | |||
+ | Clearly, B is divisible by 1 doesn't make sense, so these are the only possibilities. Summing these fractions give <math>\frac{169}{550}</math>, so the answer is 169 + 550 = 719. | ||
+ | |||
+ | ~Anonymus |
Revision as of 17:26, 16 February 2023
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