+0  
 
+1
58
5
avatar+54 

What is the largest possible median for the five number set $\{x, 2x, 3, 2, 5\}$ if $x$ can be any integer?

 Dec 29, 2020
 #1
avatar+80 
+1

\(2, 3, 5, x, 2x\) creates the largest median no matter what. \(x\) can be any integer you want it to be, but the median will always be \(5\)

 Dec 29, 2020
 #2
avatar+2197 
+4

Correct! Notice that the median must always be the middle numberm and we can already see that 2 and 3 are less than 5. So, 5 will be the middle number, no matter what x and 2x are.

 

Largest possible median is 5!

 Dec 29, 2020
 #3
avatar+114171 
+3

Assuming that x is a positive ineger

 

Let x  be  some "large positive integer "

 

The ordering of the  numbers  is

 

2, 3 , 5 , x , 2x

 

And the median is  5.....just  as   cryptoaops   found  !!!

 

 Note  that not just  ANY integer will produce this....for example...if  x  = 1   we have

1 2 2 3 5    and the median is  2  

 

And if x  = 2  we  have    2  2  3  4  5  and the median    = 3

 

If x =   3  we have   2 3 3  5  6       median  = 3

 

If x =  4  we have   2 3 4 5  8       median = 4

 

In fact  x  must be an integer greater than  4 to have a median of 5

 

cool cool cool 

 Dec 29, 2020
 #4
avatar+54 
+1

Thanks guys! :D

 Dec 29, 2020
 #5
avatar+80 
+1

No problem!

 

Also, I wanted to notify you that there should be no cheating on Alcumus or Homework as it is not allowed on AoPS.

cryptoaops  Dec 29, 2020

38 Online Users

avatar
avatar
avatar