Difference between revisions of "2017 AIME II Problems/Problem 5"

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A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are <math>189</math>, <math>320</math>, <math>287</math>, <math>234</math>, <math>x</math>, and <math>y</math>. Find the greatest possible value of <math>x+y</math>.
 
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are <math>189</math>, <math>320</math>, <math>287</math>, <math>234</math>, <math>x</math>, and <math>y</math>. Find the greatest possible value of <math>x+y</math>.
  
==Solution==
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==Solution 1==
Let these four numbers be <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math>, where <math>a>b>c>d</math>. <math>x+y</math> needs to be maximized, so let <math>x=a+b</math> and <math>y=a+c</math> because these are the two largest pairwise sums. Now <math>x+y=2a+b+c</math> needs to be maximized. Notice <math>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))</math>. No matter how the numbers <math>189</math>, <math>320</math>, <math>287</math>, and <math>234</math> are assigned to the values <math>a+d</math>, <math>b+c</math>, <math>b+d</math>, and <math>c+d</math>, the sum <math>(a+d)+(b+c)+(b+d)+(c+d)</math> will always be <math>189+320+287+234</math>. Therefore we need to maximize <math>3((a+c)+(b+d))-(189+320+287+234)</math>. The maximum value of <math>(a+c)+(b+d)</math> is achieved when we let <math>a+c</math> and <math>b+d</math> be <math>320</math> and <math>287</math> because these are the two largest pairwise sums besides <math>x</math> and <math>y</math>. Therefore, the maximum possible value of <math>x+y=3(320+287)-(189+320+287+234)=\boxed{791}</math>.
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Let these four numbers be <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math>, where <math>a>b>c>d</math>. <math>x+y</math> needs to be maximized, so let <math>x=a+b</math> and <math>y=a+c</math> because these are the two largest pairwise sums. Now <math>x+y=2a+b+c</math> needs to be maximized. Notice that <math>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))</math>. No matter how the numbers <math>189</math>, <math>320</math>, <math>287</math>, and <math>234</math> are assigned to the values <math>a+d</math>, <math>b+c</math>, <math>b+d</math>, and <math>c+d</math>, the sum <math>(a+d)+(b+c)+(b+d)+(c+d)</math> will always be <math>189+320+287+234</math>. Therefore we need to maximize <math>3((a+c)+(b+d))-(189+320+287+234)</math>. The maximum value of <math>(a+c)+(b+d)</math> is achieved when we let <math>a+c</math> and <math>b+d</math> be <math>320</math> and <math>287</math> because these are the two largest pairwise sums besides <math>x</math> and <math>y</math>. Therefore, the maximum possible value of <math>x+y=3(320+287)-(189+320+287+234)=\boxed{791}</math>.
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==Solution 2==
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Let the four numbers be <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math>, in no particular order. Adding the pairwise sums, we have <math>3a+3b+3c+3d=1030+x+y</math>, so <math>x+y=3(a+b+c+d)-1030</math>. Since we want to maximize <math>x+y</math>, we must maximize <math>a+b+c+d</math>.
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Of the four sums whose values we know, there must be two sums that add to <math>a+b+c+d</math>. To maximize this value, we choose the highest pairwise sums, <math>320</math> and <math>287</math>. Therefore, <math>a+b+c+d=320+287=607</math>.
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We can substitute this value into the earlier equation to find that <math>x+y=3(607)-1030=1821-1030=\boxed{791}</math>.
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==Solution 3==
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Note that if <math>a>b>c>d</math> are the elements of the set, then <math>a+b>a+c>b+c,a+d>b+d>c+d</math>. Thus we can assign <math>a+b=x,a+c=y,b+c=320,a+d=287,b+d=234,c+d=189</math>. Then <math>x+y=(a+b)+(a+c)=\left[(a+d)-(b+d)+(b+c)\right]+\left[(a+d)-(c+d)+(b+c)\right]=791</math>.
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==Solution 4 ( Short Casework )==
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There are two cases we can consider. Let the elements of our set be denoted <math>a,b,c,d</math>, and say that the largest sums <math>x</math> and <math>y</math> will be consisted of <math>b+d</math> and <math>c+d</math>. Thus, we want to maximize <math>b+c+2d</math>, which means <math>d</math> has to be as large as possible, and <math>a</math> has to be as small as possible to maximize <math>b</math> and <math>c</math>. So, the two cases we look at are:
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[b]Case 1:[/b]
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<cmath>a+d = 287</cmath>
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<cmath>b+c = 320</cmath>
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<cmath>a+b = 234</cmath>
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<cmath>a+c = 189</cmath>
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[b]Case 2:[/b]
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<cmath>a+d = 320</cmath>
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<cmath>a+c = 287</cmath>
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<cmath>a+b = 234</cmath>
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<cmath>b+c = 189</cmath>
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Note we have determined these cases by maximizing the value of <math>a+d</math> determined by our previous conditions. So, the answers for each ( after some simple substitution ) will be:
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[b]Case 1:[/b]
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<cmath>(a,b,c,d) = (\frac{103}{2},\frac{365}{2},\frac{275}{2},\frac{471}{2})</cmath>
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[b]Case 1:[/b]
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<cmath>(a,b,c,d) = (166,68,121,77)</cmath>
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See the first case has our largest <math>d</math>, so our answer will be <math>471+\frac{640}{2} = \boxed{791}</math>
  
 
=See Also=
 
=See Also=
{{AIME box|year=2017|n=II|num-b=4|num-a=5}}
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{{AIME box|year=2017|n=II|num-b=4|num-a=6}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 13:01, 30 April 2021

Problem

A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189$, $320$, $287$, $234$, $x$, and $y$. Find the greatest possible value of $x+y$.

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 $x+y=(a+b)+(a+c)=\left[(a+d)-(b+d)+(b+c)\right]+\left[(a+d)-(c+d)+(b+c)\right]=791$.

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:

[b]Case 1:[/b]

\[a+d = 287\] \[b+c = 320\] \[a+b = 234\] \[a+c = 189\]

[b]Case 2:[/b]

\[a+d = 320\] \[a+c = 287\] \[a+b = 234\] \[b+c = 189\]

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:

[b]Case 1:[/b]

\[(a,b,c,d) = (\frac{103}{2},\frac{365}{2},\frac{275}{2},\frac{471}{2})\]

[b]Case 1:[/b]

\[(a,b,c,d) = (166,68,121,77)\]

See the first case has our largest $d$, so our answer will be $471+\frac{640}{2} = \boxed{791}$

See Also

2017 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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