+0

0
86
1
+94

Find the number of pairs of integers (x,y) with $$0 \(\frac{1}{1-\frac{10}{x}} > 1 - \frac{5}{y}.$$

Mar 22, 2020
edited by chclt363  Mar 22, 2020
edited by chclt363  Mar 22, 2020
edited by chclt363  Mar 22, 2020
edited by chclt363  Mar 22, 2020

#1
+111330
+1

I remember this one from awhile back......one  of our really good members   [hectictar ] showed  me the solution

I believe the problem is as follows  :

1                       5

0   >         ________ > 1 -  ___

1   -10/x             y

1                              5

________     >   1   -  ___

1 -10 / x                      y

x                           5

_______ - 1  >    -  _____

x -10                       y

x  - (x -10)                5

__________   >    -    ___

x   - 10                    y

10                        5

_______   >     -    ______

x  -10                       y

And we know  0 < x < 10  because that is the only way    1 / [ 1  -10/x ]  can be less than 0.

Since  x - 10  is negative, when we multiply both sides by it, we flip the sign.

10  <     -  5  ( x -10)

___________

y

And we know  0 < y < 5  because that is the only way that  1  -  5/y   can be less than 0

So

Since  y  is positive, when we mult both sides by it, don't flip the sign.

10 y   <  - 5(x - 10)            this leads to

y  <   10 - x

________

2

y  <   (-1/2)x  +  5

So

We  have  these three inequalities

0 < x < 10

0 < y <  5

y < (-1/2)x  + 5

And the  intersection of  these  forms the lower-left triangle area , here :

https://www.desmos.com/calculator/y1vjodjezc

I'll let you  find  the integer solutions in this area    ....

THANKS TO HECTICTAR FOR THIS SOLUTION   !!!!!

Mar 22, 2020