AIME 2017 II · 第 5 题

Solution 1

Let these four numbers be $a$, $b$, $c$, and $d$, where $a>b>c>d$. $x+y$ needs to be maximized, so let $x=a+b$ and $y=a+c$ because these are the two largest pairwise sums. Now $x+y=2a+b+c$ needs to be maximized. Notice that $2a+b+c=3(a+b+c+d)-(a+2b+2c+3d)=3((a+c)+(b+d))-((a+d)+(b+c)+(b+d)+(c+d))$. No matter how the numbers $189$, $320$, $287$, and $234$ are assigned to the values $a+d$, $b+c$, $b+d$, and $c+d$, the sum $(a+d)+(b+c)+(b+d)+(c+d)$ will always be $189+320+287+234$. Therefore we need to maximize $3((a+c)+(b+d))-(189+320+287+234)$. The maximum value of $(a+c)+(b+d)$ is achieved when we let $a+c$ and $b+d$ be $320$ and $287$ because these are the two largest pairwise sums besides $x$ and $y$. Therefore, the maximum possible value of $x+y=3(320+287)-(189+320+287+234)=\boxed{791}$.

Solution 2

Let the four numbers be $a$, $b$, $c$, and $d$, in no particular order. Adding the pairwise sums, we have $3a+3b+3c+3d=1030+x+y$, so $x+y=3(a+b+c+d)-1030$. Since we want to maximize $x+y$, we must maximize $a+b+c+d$.

Of the four sums whose values we know, there must be two sums that add to $a+b+c+d$. To maximize this value, we choose the highest pairwise sums, $320$ and $287$. Therefore, $a+b+c+d=320+287=607$.

We can substitute this value into the earlier equation to find that $x+y=3(607)-1030=1821-1030=\boxed{791}$.

Solution 3

Note that if $a>b>c>d$ are the elements of the set, then $a+b>a+c>b+c,a+d>b+d>c+d$. Thus we can assign $a+b=x,a+c=y,b+c=320,a+d=287,b+d=234,c+d=189$. Then $x+y=(a+b)+(a+c)=\left[(a+d)-(b+d)+(b+c)\right]+\left[(a+d)-(c+d)+(b+c)\right]=\boxed{791}$.

Solution 4 (Short Casework)

There are two cases we can consider. Let the elements of our set be denoted $a,b,c,d$, and say that the largest sums $x$ and $y$ will be consisted of $b+d$ and $c+d$. Thus, we want to maximize $b+c+2d$, which means $d$ has to be as large as possible, and $a$ has to be as small as possible to maximize $b$ and $c$. So, the two cases we look at are:

Case 1:

$$a+d = 287$$ $$b+c = 320$$ $$a+b = 234$$ $$a+c = 189$$ Case 2:

$$a+d = 320$$ $$b+c = 189$$ $$a+b = 234$$ $$a+c = 287$$ Note we have determined these cases by maximizing the value of $a+d$ determined by our previous conditions. So, the answers for each ( after some simple substitution ) will be:

Case 1:

$$(a,b,c,d) = (\frac{103}{2},\frac{365}{2},\frac{275}{2},\frac{471}{2})$$ Case 2:

$$(a,b,c,d) = (166,68,121,154)$$ See the first case has our largest $d$, so our answer will be $471+\frac{640}{2} = \boxed{791}$