Statistics Olympiad Problem 
Given that the mean, median, range and the only mode of 200 integers are also 200. If $A$ is the largest integer among those 200 integers, find the maximum value of $A$. 

I have asked some of my friends and colleagues to solve this problem, but no one give me a light. 
Attempt:
Assuming first that all the numbers are $200$. 
To maximize $A$, but satisfies all the criterion given, we need to make $A$ ascending while descending the value of other numbers. 
Logically, 
$100, 200, 200, \cdots, 300$
still satisfies.
Maybe we have$A_{\text{max}} = 300$? 
I don't know how to approach it clearly. 
 A: If $A\gt 400$ then, since the range is $200$, all the integers would exceed $200$, making the mean exceed $200$, so that is not possible
If $A=400$ then, since the range is $200$, all the integers would be at least $200$ and at least one is strictly greater, making the mean exceed $200$, so that is not possible
If $A=399$ then, since the range is $200$, the minimum would be $199$.  Since the  median is $200$, no more than $99$ of the integers can be $199$, with mean at least $\frac{1}{200}(99 \times 199 + 100 \times 200 +1 \times 399)=200.5$, making the mean exceed $200$, so that is not possible
If $A=398$ then, since the range is $200$, the minimum would be $198$.  There is a solution with 


*

*$198$ appearing $99$ times, 

*$200$ appearing $100$ times, and 

*$398$ appearing $1$ time, 


with mean $\frac{1}{200}(99 \times 198 + 100 \times 200 +1 \times 398)=200$, and clearly the range, median and mode are all $200$ too, so that is possible
So the maximum possible value of $A$ is $398$
A: The distribution consisting of $\{196, \underbrace{198, \ldots, 198}_{96}, \underbrace{200, \ldots 200}_{102}, 396 \}$
Has mean = mode = range = $200$ and $A_{max} = 396$.
Here's the histogram, with the two "singletons" barely visible:

