Just an average problem

In order to make a single value as high as possible, the others must be as small as possible to keep the average down. Given that we are only allowed to use integers, and they must all be distinct, the best we can do is to use 1 through 9, along with our mystery number, x.

Based on this, we set up $\frac {1+2+3+4+5+6+7+8+9+x}{10}=20$ .

Solving this equation leaves us with an answer of $\boxed{155}$

we have ten distinct positive integers which have an average of 20. This means that the sum of these ten positive integers should be 200. To get the largest possible value of a single number in the list, we need to subtract the sum of the nine lowest positive integers (nine because the tenth number is the largest number we're looking for), these are 1,2,3,4,5,6,7,8, and 9 from 200. The sum of these nine lowest numbers is 45. To get the largest value or the tenth number, we subtract 45 from 200 which will give us 155.

In order for 1 number among 10 numbers to be the largest it possibly can and the sum be equal to 200, the other 9 numbers have to be as small as possible.

The numbers have to be distinct positive integers, so the set of 9 numbers that maximizes the 10th number is 1,2,..,9.

So, the sum of the first 9 numbers is 45.

45+K=200

K=155