2009 AIME I Problems/Problem 14

Problem

For $t = 1, 2, 3, 4$ , define $S_t = \sum_{i = 1}^{350}a_i^t$ , where $a_i \in \{1,2,3,4\}$ . If $S_1 = 513$ and $S_4 = 4745$ , find the minimum possible value for $S_2$ .

Solution

Because the order of the $a$ s doesn't matter, we simply need to find the number of $1$ s $2$ s $3$ s and $4$ s that minimize $S_2$ . So let $w, x, y,$ and $z$ represent the number of $1$ s, $2$ s, $3$ s, and $4$ s respectively. Then we can write three equations based on these variables. Since there are a total of $350 a$ s, we know that $w + x + y + z = 350$ . We also know that $w + 2x + 3y + 4z = 513$ and $w + 16x + 81y + 256z = 4745$ .

2009 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

Page

Toolbox

Search

2009 AIME I Problems/Problem 14

Problem

Solution

See also