2001 AMC 12 Problems

Problem 1

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?

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

Solution

Problem 2

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 $N$ ?

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

Solution

Problem 3

The state income tax where Kristin lives is levied at the rate of $p%$ (Error compiling LaTeX. Unknown error_msg) of the first <dollar/> $28000$ of annual income plus $(p + 2)%$ (Error compiling LaTeX. Unknown error_msg) of any amount above <dollar/> $28000$ . Kristin noticed that the state income tax she paid amounted to $(p + 0.25)%$ (Error compiling LaTeX. Unknown error_msg) of her annual income. What was her annual income?

$\text{(A)}\,$ <dollar/> $28000\qquad \text{(B)}\,$ <dollar/> $32000\qquad \text{(C)}\,$ <dollar/> $35000\qquad \text{(D)}\,$ <dollar/> $42000\qquad \text{(E)}\,$ <dollar/> $56000$

Solution

Problem 4

The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$ . What is their sum?