2

Part of 2001 AIME Problems

Problems(2)

Source:

\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}.

Students Studying Foreign Languages

Source:

Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let

m

be the smallest number of students who could study both languages, and let

M

be the largest number of students who could study both languages. Find

M-m

percentAMCAIMEfloor functionceiling function