2009 AIME I Problems/Problem 14

Revision as of 15:13, 20 March 2009 by Kubluck (talk | contribs) (New page: == Problem == For <math>t = 1, 2, 3, 4</math>, define <math>S_t = \sum_{i = 1}^{350}a_i^t</math>, where <math>a_i \in \{1,2,3,4\}</math>. If <math>S_1 = 513</math> and <math>S_4 = 4745</ma...)
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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 $w + 16x + 81y + 256z = 4745$.

See also

2009 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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All AIME Problems and Solutions
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