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Math Help - set problem

  1. #1
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    set problem

    Let Si be the set of all intergers n such that 100i < or = n < 100 (i + 1).
    For example, S4 is the set {400, 401,402,........499}. How many of the sets S0, S1, S2, S3,................ S999 do not contain a perfect square?

    I tried to make a pattern
    1 2 3 4 5 6 7 8 ............ 50
    1 4 9 16 25 36 49 64 ............ 2500
    3 5 7 9 11 13 15
    and I see the difference between two squares have a pattern of 2n-1

    therefore, when 2n-1 >100 there will be some sets that don't have a square. n <50 there will be squares in each set.
    Up to S25 there should be at least one square in each set.

    I am stuck from here because it will be impossible for me to try out all the sets...



    Thanks.

    Vicky.
    Last edited by Vicky1997; December 6th 2009 at 05:06 PM.
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  2. #2
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    Quote Originally Posted by Vicky1997 View Post
    Let Si be the set of all intergers n such that 100i < or = n < 100 (i + 1).
    For example, S4 is the set {400, 401,402,........499}. How many of the sets S0 S1 S2 S3,................ S999 do not contain a perfect square?
    I do not understand this question as posted.
    The fact is that each set S_i as defined contains a square.
    Are you asking "how many subsets of S_n do not contain a square?"
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  3. #3
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    Quote Originally Posted by Plato View Post
    I do not understand this question as posted.
    The fact is that each set S_i as defined contains a square.
    Are you asking "how many subsets of S_n do not contain a square?"
    I was trying to find the sets that do not contain a square.
    For example, I think S91 does not contain a square because the square of 95 is 9025 and the square of 96 is 9216.

    Vicky.
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  4. #4
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    Quote Originally Posted by Vicky1997 View Post
    Let Si be the set of all intergers n such that 100i < or = n < 100 (i + 1).
    For example, S4 is the set {400, 401,402,........499}. How many of the sets S0, S1, S2, S3,................ S999 do not contain a perfect square?

    I tried to make a pattern
    1 2 3 4 5 6 7 8 ............ 50
    1 4 9 16 25 36 49 64 ............ 2500
    3 5 7 9 11 13 15
    and I see the difference between two squares have a pattern of 2n-1

    therefore, when 2n-1 >100 there will be some sets that don't have a square. n <50 there will be squares in each set.
    Up to S25 there should be at least one square in each set.

    I am stuck from here because it will be impossible for me to try out all the sets...



    Thanks.

    Vicky.

    Of course, it's impossible to try out all the numbers. You did all the work and stopped right before you got the answer.
    As you said (n+1)squared - n sqaured >100
    so n>=50
    50 squared is 2500 and this number is in the set S25.
    S0 to S25 all have at least one square.

    Now let's look at the sets S26 to S999.
    Most of the sets will not have a square and the ones that do can't have more than one. This fact makes the answer obvious.

    The biggest number is the set is 99999 and this is in between 316 sqaured and 317 squared. This means a total of 316 squares and we already know that 50 of them are in the sets from S0 to S25.

    316 - 50 = 266
    This shows that there are 266 squares in the 974 sets from S26 to S999.
    974-266 = 708

    The answer should be 708. I went over my work 5 times to make sure there are no errors in my calculation.

    If you still haven't solved this problem,
    you have to admit I'm one step ahead of you.
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  5. #5
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    Hi Ronsy,

    You did a good job. But I solved the problem last year.(you are not ahead of me yet)

    Vicky.
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