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

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==Solution 1==
 
==Solution 1==
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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 <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>.
  
 
==Solution 2==
 
==Solution 2==

Revision as of 11:32, 4 June 2020

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 $x+y=(a+b)+(a+c)=2((a+d)+(b+c))-((c+d)+(b+d))=791$.

See Also

2017 AIME II (Problems • Answer Key • Resources)
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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