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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+117 
+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 
+1

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+117546 
+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+82 
+1

Thanks guys! :D

 Dec 29, 2020
 #5
avatar+117 
+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

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