2001 AIME I Problems/Problem 2

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, $\frac{ax+1}{a+1}=x-13$ and $\frac{ax+2001}{a+1}=x+27$ $\frac{ax+2001}{a+1}-40=\frac{ax+1}{a+1}$ $\frac{2000}{a+1}=40$ so $2000=40(a+1)$ $a=49$ We plug that into our very first formula, and get: $\frac{49x+1}{50}=x-13$ $49x+1=50x-650$ $x=\boxed{651}$

2001 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2001 AIME I Problems/Problem 2

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