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    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]
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    (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?
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    Interval is (0,1), since 0,1\notin S
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    (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...
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    (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].
 
 
 
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