2003 AIME I Problems/Problem 3

Problem

Let the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}.$ Susan makes a list as follows: for each two-element subset of $\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.

Solution

Order the numbers in the set from greatest to least to reduce error: $\{34, 21, 13, 8, 5, 3, 2, 1\}.$ Each element of the set will appear in $7$ two-element subsets, once with each other number.

$34$ will be the greater number in $7$ subsets.
$21$ will be the greater number in $6$ subsets.
$13$ will be the greater number in $5$ subsets.
$8$ will be the greater number in $4$ subsets.
$5$ will be the greater number in $3$ subsets.
$3$ will be the greater number in $2$ subsets.
$2$ will be the greater number in $1$ subsets.
$1$ will be the greater number in $0$ subsets.

Therefore the desired sum is $34\cdot7+21\cdot6+13\cdot5+8\cdot4+5\cdot3+3 \cdot2+2\cdot1+1\cdot0=\boxed{484}$ .

2003 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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2003 AIME I Problems/Problem 3

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