You are Here: Home >< Maths

# Help with BMO question on rationality? watch

1. Hi,

I'm having trouble getting the answer to this BMO question in an appropriately rigorous form...

Let S be the set of rational numbers with the following properties:
1) 1/2 is an element of S
2) If x is an element of S, then so is 1/x+1 and x/x+1.

Prove that S contains all rational numbers in the interval [0,1]
2. (Original post by MaffsIsFun)
Hi,

I'm having trouble getting the answer to this BMO question in an appropriately rigorous form...

Let S be the set of rational numbers with the following properties:
1) 1/2 is an element of S
2) If x is an element of S, then so is 1/x+1 and x/x+1.

Prove that S contains all rational numbers in the interval [0,1]
What have you tried?
3. Interval is (0,1), since
4. (Original post by RDKGames)
What have you tried?
I've managed to show that if a/b is some rational number in S then this implies that 1/2 is, but unfortunately this argument works the wrong way...
5. (Original post by MaffsIsFun)
I've managed to show that if a/b is some rational number in S then this implies that 1/2 is, but unfortunately this argument works the wrong way...
Well, one method that I think *will* work.

Suppose you want to show p/q is in S.

If you could find x in S s.t. 1/(x+1) = p/q

or alternatively x in S s.t. x/(x+1) = p/q

then you'd be done.

If you solve for x, it shouldn't be that hard to decide on a suitable induction hypothesis that will let you claim x *is* in S.

[A little fiddling around starting from 1/2 to see what numbers you can easily form will probably also help with an induction hypothesis].

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: November 9, 2016
The home of Results and Clearing

### 1,023

people online now

### 1,567,000

students helped last year
Today on TSR

### University open days

1. Sheffield Hallam University
Tue, 21 Aug '18
2. Bournemouth University
Wed, 22 Aug '18
3. University of Buckingham
Thu, 23 Aug '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams