2002 AMC 10B

What is the value of the ratio

$$
\dfrac{2^{2001} \cdot 3^{2003}}{6^{2002}}?
$$

For the nonzero numbers $a,$ $b,$ and $c,$ define $$
(a, b, c) = \dfrac{abc}{a+b+c}.
$$ What is $(2, 4, 6)?$

The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M,$ all of whose digits are distinct. Which digit does the number $M$ not contain?

What is the value of $$
\begin{aligned} &(3x - 2)(4x + 1) \\ &\quad {}- (3x - 2)4x + 1 \end{aligned}
$$ when $x = 4?$

Circles of radius $2$ and $3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. What is the area of the shaded region?

For how many positive integers $n$ is $n^2 - 3n + 2$ a prime number?

Let $n$ be a positive integer such that $\dfrac12 + \dfrac13 + \dfrac17 + \dfrac1n$ is an integer. Which of the following statements is **not** true?

Suppose July of year $N$ has five Mondays. Which of the following must occur five times in August of year $N?$ (Note: both months have $31$ days.)

Using the letters $A,$ $M,$ $O,$ $S,$ and $U,$ we can form $120$ five-letter “words.” If these “words” are arranged in alphabetical order, then the “word” $USAMO$ occupies which position?

Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2 + ax + b = 0$ has solutions $a$ and $b.$ What is the pair $(a, b)?$

The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares?

For which of the following values of $k$ does the equation $$
\dfrac{x - 1}{x - 2} = \dfrac{x - k}{x - 6}
$$ have no solution for $x?$

What value of $x$ makes $8xy - 12y + 2x - 3 = 0$ true for all values of $y?$

The number $25^{64} \cdot 64^{25}$ is the square of a positive integer $N.$ In decimal representation, what is the sum of the digits of $N?$

The positive integers $A,$ $B,$ $A - B,$ and $A + B$ are all prime numbers. The sum of these four primes is

For how many integers $n$ is $\dfrac{n}{20 - n}$ the square of an integer?

A regular octagon $ABCDEFGH$ has sides of length two. What is the area of $\triangle ADG?$

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

Suppose that $\{a_n\}$ is an arithmetic sequence with $$
a_1 + a_2 + \cdots + a_{100} = 100
$$ and $$
a_{101} + a_{102} + \cdots + a_{200} = 200.
$$ What is the value of $a_2 - a_1?$

Let $a,$ $b,$ and $c$ be real numbers such that $a - 7b + 8c = 4$ and $8a + 4b - c = 7.$ What is $a^2 - b^2 + c^2?$