Sum of fractions with restricted domain and range

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Find all positive integers $m$ and $n$ so that for any $x$ and $y$ in the interval $[m, n]$, the value of $5/x + 7/y$ will also be in $ [m, n]$.



I evaluated the inequalities into



$(ym-5)(xm-7) < 35$



$(yn-5)(xn-7) > 35$



but can't really think of what to do next.




inequality




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  • Are $x,y$ assumed to be positive?
    – Dr. Sonnhard Graubner
    Jul 31 at 10:16










  • yes they are000
    – SuperMage1
    Jul 31 at 10:17










  • You need $12nle n$, which is a bit difficult for non-zero numbers. Where did you get this question?
    – Macavity
    Jul 31 at 12:38










  • pmo.ph/wp-content/uploads/2014/08/16th-PMO-Area.pdf
    – SuperMage1
    Jul 31 at 13:02










  • Its a bit early into the test (number 6), O
    – SuperMage1
    Jul 31 at 13:03















up vote
0
down vote

favorite












Find all positive integers $m$ and $n$ so that for any $x$ and $y$ in the interval $[m, n]$, the value of $5/x + 7/y$ will also be in $ [m, n]$.



I evaluated the inequalities into



$(ym-5)(xm-7) < 35$



$(yn-5)(xn-7) > 35$



but can't really think of what to do next.




inequality




share|cite|improve this question





















  • Are $x,y$ assumed to be positive?
    – Dr. Sonnhard Graubner
    Jul 31 at 10:16










  • yes they are000
    – SuperMage1
    Jul 31 at 10:17










  • You need $12nle n$, which is a bit difficult for non-zero numbers. Where did you get this question?
    – Macavity
    Jul 31 at 12:38










  • pmo.ph/wp-content/uploads/2014/08/16th-PMO-Area.pdf
    – SuperMage1
    Jul 31 at 13:02










  • Its a bit early into the test (number 6), O
    – SuperMage1
    Jul 31 at 13:03













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Find all positive integers $m$ and $n$ so that for any $x$ and $y$ in the interval $[m, n]$, the value of $5/x + 7/y$ will also be in $ [m, n]$.



I evaluated the inequalities into



$(ym-5)(xm-7) < 35$



$(yn-5)(xn-7) > 35$



but can't really think of what to do next.




inequality




share|cite|improve this question













Find all positive integers $m$ and $n$ so that for any $x$ and $y$ in the interval $[m, n]$, the value of $5/x + 7/y$ will also be in $ [m, n]$.



I evaluated the inequalities into



$(ym-5)(xm-7) < 35$



$(yn-5)(xn-7) > 35$



but can't really think of what to do next.





inequality





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 31 at 13:04
























asked Jul 31 at 10:14









SuperMage1

56719




56719











  • Are $x,y$ assumed to be positive?
    – Dr. Sonnhard Graubner
    Jul 31 at 10:16










  • yes they are000
    – SuperMage1
    Jul 31 at 10:17










  • You need $12nle n$, which is a bit difficult for non-zero numbers. Where did you get this question?
    – Macavity
    Jul 31 at 12:38










  • pmo.ph/wp-content/uploads/2014/08/16th-PMO-Area.pdf
    – SuperMage1
    Jul 31 at 13:02










  • Its a bit early into the test (number 6), O
    – SuperMage1
    Jul 31 at 13:03

















  • Are $x,y$ assumed to be positive?
    – Dr. Sonnhard Graubner
    Jul 31 at 10:16










  • yes they are000
    – SuperMage1
    Jul 31 at 10:17










  • You need $12nle n$, which is a bit difficult for non-zero numbers. Where did you get this question?
    – Macavity
    Jul 31 at 12:38










  • pmo.ph/wp-content/uploads/2014/08/16th-PMO-Area.pdf
    – SuperMage1
    Jul 31 at 13:02










  • Its a bit early into the test (number 6), O
    – SuperMage1
    Jul 31 at 13:03
















Are $x,y$ assumed to be positive?
– Dr. Sonnhard Graubner
Jul 31 at 10:16




Are $x,y$ assumed to be positive?
– Dr. Sonnhard Graubner
Jul 31 at 10:16












yes they are000
– SuperMage1
Jul 31 at 10:17




yes they are000
– SuperMage1
Jul 31 at 10:17












You need $12nle n$, which is a bit difficult for non-zero numbers. Where did you get this question?
– Macavity
Jul 31 at 12:38




You need $12nle n$, which is a bit difficult for non-zero numbers. Where did you get this question?
– Macavity
Jul 31 at 12:38












pmo.ph/wp-content/uploads/2014/08/16th-PMO-Area.pdf
– SuperMage1
Jul 31 at 13:02




pmo.ph/wp-content/uploads/2014/08/16th-PMO-Area.pdf
– SuperMage1
Jul 31 at 13:02












Its a bit early into the test (number 6), O
– SuperMage1
Jul 31 at 13:03





Its a bit early into the test (number 6), O
– SuperMage1
Jul 31 at 13:03











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Hint You need $mle 12/n$ and $nge 12/m$ so $mn=12$, find the possibilities for positive integers $m<n$…






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    active

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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote



    accepted










    Hint You need $mle 12/n$ and $nge 12/m$ so $mn=12$, find the possibilities for positive integers $m<n$…






    share|cite|improve this answer

























      up vote
      1
      down vote



      accepted










      Hint You need $mle 12/n$ and $nge 12/m$ so $mn=12$, find the possibilities for positive integers $m<n$…






      share|cite|improve this answer























        up vote
        1
        down vote



        accepted







        up vote
        1
        down vote



        accepted






        Hint You need $mle 12/n$ and $nge 12/m$ so $mn=12$, find the possibilities for positive integers $m<n$…






        share|cite|improve this answer













        Hint You need $mle 12/n$ and $nge 12/m$ so $mn=12$, find the possibilities for positive integers $m<n$…







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered Jul 31 at 14:13









        Macavity

        34.4k52351




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