Competition · AMC preparation · step 4 of 4

AMC 12 2002B: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 12 2002B #1 grade 4+ arithmetic
The arithmetic mean of the nine numbers in the set {9, 99, 999, 9999, …, 999999999} is a 9-digit number M, all of whose…
AMC 12 2002B #2 grade 6+ arithmetic
What is the value of (3x - 2)(4x + 1) - (3x - 2)4x + 1 when x=4?
AMC 12 2002B #3 grade 6+ number-theory
For how many positive integers n is n² - 3n + 2 a prime number?
AMC 12 2002B #4 grade 5+ arithmetic
Let n be a positive integer such that 1/2 + 1/3 + 1/7 + 1/n is an integer. Which of the following statements is not true…
AMC 12 2002B #5 grade 8+ algebra, geometry-2d
Let v, w, x, y, and z be the degree measures of the five angles of a pentagon. Suppose that v < w < x < y < z and v, w,…
AMC 12 2002B #6 grade 8+ algebra
Suppose that a and b are nonzero real numbers, and that the equation x² + ax + b = 0 has solutions a and b. Then the pai…
AMC 12 2002B #7 grade 8+ arithmetic
The product of three consecutive positive integers is 8 times their sum. What is the sum of their squares?
AMC 12 2002B #8 grade 4+ arithmetic
Suppose July of year N has five Mondays. Which of the following must occur five times in the August of year N? (Note: Bo…
AMC 12 2002B #9 grade 9+ algebra
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…
AMC 12 2002B #10 grade 7+ counting
How many different integers can be expressed as the sum of three distinct members of the set {1,4,7,10,13,16,19}?
AMC 12 2002B #11 grade 4+ number-theory
The positive integers A, B, A-B, and A+B are all prime numbers. The sum of these four primes is
AMC 12 2002B #12 grade 8+ geometry-2d
For how many integers n is n/(20-n) the square of an integer?
AMC 12 2002B #13 grade 8+ number-theory
The sum of 18 consecutive positive integers is a perfect square. The smallest possible value of this sum is
AMC 12 2002B #14 grade 7+ arithmetic
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles inter…
AMC 12 2002B #15 grade 7+ number-theory
How many four-digit numbers N have the property that the three-digit number obtained by removing the leftmost digit is o…
AMC 12 2002B #16 grade 7+ probability
Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Then Amal rolls a fair six-sided die. What…
AMC 12 2002B #17 grade 6+ geometry-2d
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…
AMC 12 2002B #18 grade 9+ probability
A point P is randomly selected from the rectangular region with vertices (0,0),(2,0),(2,1),(0,1). What is the probabilit…
AMC 12 2002B #19 grade 8+ algebra
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
AMC 12 2002B #20 grade 8+ geometry-2d
Let △ XOY be a right-angled triangle with m∠ XOY = 90°. Let M and N be the midpoints of legs OX and OY, respectively. Gi…
AMC 12 2002B #21 grade 6+ number-theory
For all positive integers n less than 2002, let a_n ={ 11, if n is divisible by 13 and 14; 13, if n is divisible by 14 a…
AMC 12 2002B #22 grade 11+ algebra
For all integers n greater than 1, define a_n = 1/(log_n 2002). Let b = a₂ + a₃ + a₄ + a₅ and c = a₁₀ + a₁₁ + a₁₂ + a₁₃…
AMC 12 2002B #23 grade 8+ geometry-2d
In △ ABC, we have AB = 1 and AC = 2. Side BC and the median from A to BC have the same length. What is BC?
AMC 12 2002B #24 grade 8+ geometry-2d
A convex quadrilateral ABCD with area 2002 contains a point P in its interior such that PA = 24, PB = 32, PC = 28, PD =…
AMC 12 2002B #25 grade 9+ geometry-2d
Let f(x) = x² + 6x + 1, and let R denote the set of points (x,y) in the coordinate plane such that f(x) + f(y) ≤ 0 and f…

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