2001 AMC 10 Problems

Problem 1

The median of the list $n, n + 3, n + 4, n + 5, n + 6, n + 8, n + 10, n + 12, n + 15$ is $10$ . What is the mean?

$\mathrm{(A)}\ 4 \qquad\mathrm{(B)}\ 6 \qquad\mathrm{(C)}\ 7 \qquad\mathrm{(D)}\ 10 \qquad\mathrm{(E)}\ 11$

Problem 2

A number $x$ is $2$ more than the product of its reciprocal and its additive inverse. In which interval does the number lie?

$\mathrm{(A)}\ -4\leq x\leq -2 \qquad\mathrm{(B)}\ -2<x\leq 0 \qquad\mathrm{(C)}\ 0<x\leq 2$

$\mathrm{(D)}\ 2<x\leq 4 \qquad\mathrm{(E)}\ 4<x\leq 6$

Problem 3

The sum of two numbers is $S$ . Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

$\mathrm{(A)}\ 2S+3 \qquad\mathrm{(B)}\ 3S+2 \qquad\mathrm{(C)}\ 3S+6 \qquad\mathrm{(D)}\ 2S+6 \qquad\mathrm{(E)}\ 2S+12$

Problem 4

What is the maximum number for the possible points of intersection of a circle and a triangle?

$\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 3 \qquad\mathrm{(C)}\ 4 \qquad\mathrm{(D)}\ 5 \qquad\mathrm{(E)}\ 6$

Problem 5

How many of the twelve pentominoes pictured below have at least one line of symmetry?

$\mathrm{(A)}\ 3 \qquad\mathrm{(B)}\ 2 \qquad\mathrm{(C)}\ 5 \qquad\mathrm{(D)}\ 6 \qquad\mathrm{(E)}\ 7$

Problem 6

Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$ . For example, $P(23) = 6$ and $S(23) = 5$ . Suppose $N$ is a two-digit number such that $N = P(N)+S(N)$ . What is the units digit of $N$ ?

$\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 3 \qquad\mathrm{(C)}\ 6 \qquad\mathrm{(D)}\ 8 \qquad\mathrm{(E)}\ 9$

Problem 7

When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?

$\mathrm{(A)}\ 0.0002 \qquad\mathrm{(B)}\ 0.002 \qquad\mathrm{(C)}\ 0.02 \qquad\mathrm{(D)}\ 0.2 \qquad\mathrm{(E)}\ 2$

Problem 8

Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?

$\mathrm{(A)}\ 42 \qquad\mathrm{(B)}\ 84 \qquad\mathrm{(C)}\ 126 \qquad\mathrm{(D)}\ 178 \qquad\mathrm{(E)}\ 252$

Solutions

1. The median is $n+6$ , therefore $n=4$ . Computation shows that the sum of all numbers is $99$ and thus the mean is $99/9=11$ .

2. The reciprocal of $x$ is $\frac 1x$ and the additive inverse is $-x$ . (Note that $x$ must be non-zero to have a reciprocal.) The product of these two is $\frac 1x \cdot (-x) = -1$ . Thus $x$ is $2$ more than $-1$ . Therefore $x=1$ .

3. The original two numbers are $x$ and $y$ , with $x+y=S$ . The new two numbers are $2(x+3)$ and $2(y+3)$ . Their sum is $2(x+3)+2(y+3)=2x+2y+12=2(x+y)+12 = 2S+12$ .

4. Each side of the triangle can only intersect the circle twice, so the maximum is at most 6. This can be achieved: $[asy] unitsize(0.3cm); draw( circle((0,-0.2),2.2) ); draw( (-2,-2)--(2,-2)--(0,3)--cycle ); [/asy]$

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