2002 AMC 12A

Compute the sum of all the roots of $$
\begin{aligned} &(2x+3)(x-4) \\ &\quad {}+(2x+3)(x-6)=0. \end{aligned}
$$

Cindy was asked by her teacher to subtract $3$ from a certain number and then divide the result by $9.$ Instead, she subtracted $9$ and then divided the result by $3,$ giving an answer of $43.$ What would her answer have been had she worked the problem correctly?

According to the standard convention for exponentiation, $$
2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{16} = 65{,}536.
$$ If the order in which the exponentiations are performed is changed, how many other values are possible?

Find the degree measure of an angle whose complement is $25\%$ of its supplement.

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

For how many positive integers $m$ does there exist at least one positive integer $n$ such that $m \cdot n \le m + n?$

If an arc of $45^\circ$ on circle $A$ has the same length as an arc of $30^\circ$ on circle $B,$ then the ratio of the area of circle $A$ to the area of circle $B$ is

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $R$ the area of the red square. Which of the following is correct?

Jamal wants to store $30$ computer files on floppy disks, each of which has a capacity of $1.44$ megabytes (mb). Three of his files require $0.8$ mb of memory each, $12$ more require $0.7$ mb each, and the remaining $15$ require $0.4$ mb each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?

Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?

Mr. Earl E. Bird leaves his house for work at exactly $8{:}00$ A.M. every morning. When he averages $40$ miles per hour, he arrives at his workplace three minutes late. When he averages $60$ miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?

Both roots of the quadratic equation $$
x^2 - 63x + k = 0
$$ are prime numbers. The number of possible values of $k$ is

Two different positive numbers $a$ and $b$ each differ from their reciprocals by $1.$ What is $a + b?$

For all positive integers $n,$ let $f(n) = \log_{2002} n^2.$ Let $$
N = f(11) + f(13) + f(14).
$$ Which of the following relations is true?

The mean, median, unique mode, and range of a collection of eight integers are all equal to $8.$ The largest integer that can be an element of this collection is

Tina randomly selects two distinct numbers from the set $\{1, 2, 3, 4, 5\},$ and Sergio randomly selects a number from the set $\{1, 2, \ldots, 10\}.$ The probability that Sergio’s number is larger than the sum of the two numbers chosen by Tina is

Several sets of prime numbers, such as $\{7, 83, 421, 659\},$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?

Let $C_1$ and $C_2$ be circles defined by $$
(x - 10)^2 + y^2 = 36
$$ and $$
(x + 15)^2 + y^2 = 81,
$$ respectively. What is the length of the shortest line segment $\overline{PQ}$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q?$

The graph of the function $f$ is shown below. How many solutions does the equation $f(f(x)) = 6$ have?

Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?