MHB Find the smallest possible value of a fraction

[anemone]

Let $x,\,y,\,z$ be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallest possible value of $\dfrac{xy+z}{x+y+z}$.

 

[Albert1]

Let $x,\,y,\,z$ be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallest possible value of $\dfrac{xy+z}{x+y+z}$.

Spoiler

the smallest possible value of $\dfrac{xy+z}{x+y+z}$.will exist when xy<x+y
if x=1 then y=1,2,3,-----2011
if y=1 then x=1,2,3,-----2011
for $\dfrac {n}{n+1}<\dfrac {n+1}{n+2}$
for all $n\in N$
$\therefore$ the smallest value of $\dfrac{xy+z}{x+y+z}$=$\dfrac{2}{3}$
here $x=y=z=1$

 

[kaliprasad]

Spoiler

$\frac{xy+z}{x+y+z}$
= $1+ \frac{xy-x - y}{x+y+z}$
= $1+ \frac{(x-1)(y-1) -1}{x+y+z}$

(x-1)(y-1) - 1 is positive for all x and y except for x=1 or y=1 ( in the condition x, y <= 2011)

so x =1 , and y = 1

so we get given expression
= $1- \frac{1}{2+z}$
z = 1 shall make it lowest

so x = 1 = y = z shall give the value $\frac{2}{3}$