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The set S contains some real numbers, according to the following three rules.

(i) $\frac{1}{1}$ is in S

(ii) If $\frac{a}{b}$ is in S, where $\frac{a}{b}$ is written in lowest terms (that is, a and b have highest common factor 1), then $\frac{b}{2a}$ is in S.

(iii) If $\frac{a}{b}$ and $\frac{c}{d}$ are in S, where they are written in lowest terms, then $\frac{a+b}{c+d}$ is in S.

These rules are exhaustive: if these rules do not imply that a number is in S, then that number is not in S. Can you describe which numbers are in S?

Mauro ALLEGRANZA
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Aadhavan Srinivasan
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  • I am not able to proceed any further – Aadhavan Srinivasan Mar 19 '19 at 15:20
  • You must have some ideas. Can you list infinitely many numbers in the set? Can you list some real numbers that are definitely not in the set? – saulspatz Mar 19 '19 at 15:24
  • The numbers that are definitely not in the set: 1/3, ¼, 1/5, 1/6, … 2/5, 2/6, 2/7, 2/8,… 3/7, 3/8, 3/9,… 4/9, 4/10, 4/11,… 5/11, 5/12, 5/13… – Aadhavan Srinivasan Mar 19 '19 at 15:36
  • All the numbers you've listed as definitely not in the set are less than $\frac12.$ Are you willing to make a hypothesis? Can you prove it? – saulspatz Mar 19 '19 at 15:38
  • Can we say that the numbers in the set are 1/2 <= x <= 1 because improper fractions cannot be included in the set – Aadhavan Srinivasan Mar 19 '19 at 15:42
  • Certainly not the irrational numbers. I would guess that the answer is $\mathbb{Q}\cap [\frac12, 1]$ but I haven't proved it. It should be easy to prove there are no other numbers in the set. Then you have to prove that every number in the set can be constructed. Have a look at the Farey sequence – saulspatz Mar 19 '19 at 15:45
  • How bizarre. This is question 7 from PROMYS Europe 2020 Application Problem Set, https://www.maths.ox.ac.uk/system/files/attachments/Application_problems_2020.pdf – how did OP have it in March 2019? – Gerry Myerson Mar 03 '20 at 05:08
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    @GerryMyerson PROMYS reuses, reuses, reuses, reuses, reuses, reuses... this problem. It's still in PROMYS and PROMYS Europe and PROMYS India and presumably all of the PROMYS stuff. – Number Basher Feb 15 '23 at 12:40

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