2002 AMC 12B

All 25 problems from the 2002 AMC 12B. 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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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. The number $M$ does not contain the digit
Problem 2
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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 3
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For how many positive integers $n$ is $n^2-3n+2$ a prime number?
Problem 4
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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 5
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Let $v,$ $w,$ $x,$ $y,$ and $z$ be the degree measures of the five angles of a pentagon. Suppose $v\lt w\lt x\lt y\lt z$ and $v,$ $w,$ $x,$ $y,$ $z$ form an arithmetic sequence. Find the value of $x.$
Problem 6
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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 $b.$ Then the pair $(a,b)$ is
Problem 7
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The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares?
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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If $a,$ $b,$ $c,$ $d$ are positive real numbers such that $a,$ $b,$ $c,$ $d$ form an increasing arithmetic sequence and $a,$ $b,$ $d$ form a geometric sequence, then $\dfrac{a}{d}$ is
Problem 10
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How many different integers can be expressed as the sum of three distinct members of the set $\{1,4,7,10,13,16,19\}?$
Problem 11
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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 12
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For how many integers $n$ is $\dfrac{n}{20-n}$ the square of an integer?
Problem 13
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The sum of $18$ consecutive positive integers is a perfect square. The smallest possible value of this sum is
Problem 14
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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 15
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How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N?$
Problem 16
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Juan rolls a fair regular octahedral die marked with the numbers $1$ through $8.$ Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of $3?$
Problem 17
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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 18
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A point $P$ is randomly selected from the rectangular region with vertices $(0,0),$ $(2,0),$ $(2,1),$ $(0,1).$ What is the probability that $P$ is closer to the origin than it is to the point $(3,1)?$
Problem 19
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If $a,$ $b,$ and $c$ are positive real numbers such that $a(b+c)=152,$ $b(c+a)=162,$ and $c(a+b)=170,$ then $abc$ is
Problem 20
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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 $YM=22,$ find $XY.$
Problem 21
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For all positive integers $n$ less than $2002,$ let $a_n=\begin{cases} 11, & \text{if } n \text{ is divisible by } 13 \text{ and } 14;\\ 13, & \text{if } n \text{ is divisible by } 14 \text{ and } 11;\\ 14, & \text{if } n \text{ is divisible by } 11 \text{ and } 13;\\ 0, & \text{otherwise}. \end{cases}$ Calculate $\displaystyle\sum_{n=1}^{2001} a_n.$
Problem 22
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For all integers $n$ greater than $1,$ define $a_n=\dfrac{1}{\log_n 2002}.$ Let $b=a_2+a_3+a_4+a_5$ and $c=a_{10}+a_{11}+a_{12}+a_{13}+a_{14}.$ Then $b-c$ equals
Problem 23
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In $\triangle ABC,$ we have $AB=1$ and $AC=2.$ Side $BC$ and the median from $A$ to $BC$ have the same length. What is $BC?$
Problem 24
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A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA=24,$ $PB=32,$ $PC=28,$ and $PD=45.$ Find the perimeter of $ABCD.$
Problem 25
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Let $f(x)=x^2+6x+1,$ and let $R$ denote the set of points $(x,y)$ in the coordinate plane such that $f(x)+f(y)\le0$ and $f(x)-f(y)\le0.$ The area of $R$ is closest to

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 12 archive.

2002 AMC 12B

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