Difference between revisions of "2001 AIME I Problems/Problem 2"

Revision as of 18:51, 4 July 2013

Problem

A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S}\cup\{1\}$ is $13$ less than the mean of $\mathcal{S}$ , and the mean of $\mathcal{S}\cup\{2001\}$ is $27$ more than the mean of $\mathcal{S}$ . Find the mean of $\mathcal{S}$ .

Solution

Let $x$ be the mean of $\mathcal{S}$ . Let $a$ be the number of elements in $\mathcal{S}$ . Then, the given tells us that $\frac{ax+1}{a+1}=x-13$ and $\frac{ax+2001}{a+1}=x+27$ . Subtracting, we have

$\begin{align*}\frac{ax+2001}{a+1}-40=\frac{ax+1}{a+1} \Longrightarrow \frac{2000}{a+1}=40 \Longrightarrow a=49$ (Error compiling LaTeX. Unknown error_msg)

We plug that into our very first formula, and get:

$\begin{align*}\frac{49x+1}{50}&=x-13 \\

49x+1&=50x-650 \\

x&=\boxed{651}.\end{align*}$ (Error compiling LaTeX. Unknown error_msg)

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Difference between revisions of "2001 AIME I Problems/Problem 2"

Revision as of 18:51, 4 July 2013

Problem

Solution

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