2002 AMC 10B

All 25 problems from the 2002 AMC 10B. Take it as a timed mock test — 1:15:00 on the clock, scored the way the real contest is — or read through the problems and open any of them for hints and a worked solution.

25 problems · 6 points correct, 1.5 for blank, 0 for wrong

Problem 1
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What is the value of the ratio $\dfrac{2^{2001} \cdot 3^{2003}}{6^{2002}}?$
Problem 2
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For the nonzero numbers $a,$ $b,$ and $c,$ define $(a, b, c) = \dfrac{abc}{a+b+c}.$ What is $(2, 4, 6)?$
Problem 3
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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?
Problem 4
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What is the value of $\begin{aligned} &(3x - 2)(4x + 1) \\ &\quad {}- (3x - 2)4x + 1 \end{aligned}$ when $x = 4?$
Problem 5
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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?
Problem 6
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For how many positive integers $n$ is $n^2 - 3n + 2$ a prime number?
Problem 7
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Let $n$ be a positive integer such that $\dfrac12 + \dfrac13 + \dfrac17 + \dfrac1n$ is an integer. Which of the following statements is not true?
Problem 8
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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.)
Problem 9
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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?
Problem 10
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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 $(a, b)?$
Problem 11
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The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares?
Problem 12
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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 $x?$
Problem 13
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What value of $x$ makes $8xy - 12y + 2x - 3 = 0$ true for all values of $y?$
Problem 14
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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 $N?$
Problem 15
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The positive integers $A,$ $B,$ $A - B,$ and $A + B$ are all prime numbers. The sum of these four primes is
Problem 16
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For how many integers $n$ is $\dfrac{n}{20 - n}$ the square of an integer?
Problem 17
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A regular octagon $ABCDEFGH$ has sides of length two. What is the area of $\triangle ADG?$
Problem 18
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Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
Problem 19
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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 $a_2 - a_1?$
Problem 20
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Let $a,$ $b,$ and $c$ be real numbers such that $a - 7b + 8c = 4$ and $8a + 4b - c = 7.$ What is $a^2 - b^2 + c^2?$
Problem 21
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Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and one third as fast as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first?
Problem 22
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Let $\triangle XOY$ be a right-angled triangle with $m\angle XOY = 90^\circ.$ Let $M$ and $N$ be the midpoints of legs $OX$ and $OY,$ respectively. Given that $XN = 19$ and $YM = 22,$ what is $XY?$
Problem 23
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Let $\{a_k\}$ be a sequence of integers such that $a_1 = 1$ and $a_{m+n} = a_m + a_n + mn$ for all positive integers $m$ and $n.$ What is $a_{12}?$
Problem 24
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Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $10$ vertical feet above the bottom?
Problem 25
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When $15$ is appended to a list of integers, the mean is increased by $2.$ When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1.$ How many integers were in the original list?

0 of 25 answered. Each problem left blank still scores 1.5 points.

Practise one problem at a time, with hints and solutions, on the practice page. Or work through the same ideas across every year by topic, or browse the full AMC 10 archive.

2002 AMC 10B

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25