Sum of fractions with restricted domain and range

Clash Royale CLAN TAG#URR8PPP

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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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edited Jul 31 at 13:04

asked Jul 31 at 10:14

SuperMage1

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
  
  

  
  
  
  

 | 
show 1 more comment

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

edited Jul 31 at 13:04

asked Jul 31 at 10:14

SuperMage1

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
  
  

  
  
  
  

 | 
show 1 more comment

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

edited Jul 31 at 13:04

asked Jul 31 at 10:14

SuperMage1

56719

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

edited Jul 31 at 13:04

asked Jul 31 at 10:14

SuperMage1

56719

share|cite|improve this question

share|cite|improve this question

edited Jul 31 at 13:04

edited Jul 31 at 13:04

edited Jul 31 at 13:04

asked Jul 31 at 10:14

SuperMage1

56719

asked Jul 31 at 10:14

SuperMage1

56719

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
  
  

  
  
  
  

 | 
show 1 more comment

• 
  
  
  

  
  

  
  
  
  
  
  

  
  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

 | 
show 1 more comment

1 Answer
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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

answered Jul 31 at 14:13

Macavity

34.4k52351

add a comment | 

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1 Answer
1

active

oldest

votes

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

answered Jul 31 at 14:13

Macavity

34.4k52351

add a comment | 

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

answered Jul 31 at 14:13

Macavity

34.4k52351

add a comment | 

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

answered Jul 31 at 14:13

Macavity

34.4k52351

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

answered Jul 31 at 14:13

Macavity

34.4k52351

share|cite|improve this answer

share|cite|improve this answer

answered Jul 31 at 14:13

Macavity

34.4k52351

answered Jul 31 at 14:13

Macavity

34.4k52351

answered Jul 31 at 14:13

Macavity

34.4k52351

34.4k52351

add a comment | 

add a comment | 

 

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