Competition · AMC preparation · step 4 of 4

AMC 10 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 10 2002B #1 grade 8+ rate-ratio
The ratio (2²⁰⁰¹·3²⁰⁰³)/6²⁰⁰² is:
AMC 10 2002B #2 grade 6+ arithmetic
For the nonzero numbers a, b, and c, define D(a,b,c)=abc/(a+b+c) Find D(2,4,6).
AMC 10 2002B #3 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 10 2002B #4 grade 6+ arithmetic
What is the value of (3x - 2)(4x + 1) - (3x - 2)4x + 1 when x=4?
AMC 10 2002B #5 grade 7+ geometry-2d
Circles of radius 2 and 3 are externally tangent and are circumscribed by a third circle, as shown in the figure. Find t…
AMC 10 2002B #6 grade 6+ number-theory
For how many positive integers n is n² - 3n + 2 a prime number?
AMC 10 2002B #7 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 10 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 10 2002B #9 grade 7+ arithmetic
Using the letters A, M, O, S, and U, we can form five-letter "words". If these "words" are arranged in alphabetical orde…
AMC 10 2002B #10 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 10 2002B #11 grade 8+ arithmetic
The product of three consecutive positive integers is 8 times their sum. What is the sum of their squares?
AMC 10 2002B #12 grade 8+ algebra
For which of the following values of k does the equation (x-1)/(x-2) = (x-k)/(x-6) have no solution for x?
AMC 10 2002B #13 grade 8+ arithmetic
Find the value(s) of x such that 8xy - 12y + 2x - 3 = 0 is true for all values of y.
AMC 10 2002B #14 grade 8+ number-theory
The number 25⁶⁴· 64²⁵ is the square of a positive integer N. In decimal representation, the sum of the digits of N is
AMC 10 2002B #15 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 10 2002B #16 grade 8+ number-theory
For how many integers n is n/(20-n) the square of an integer?
AMC 10 2002B #17 grade 8+ geometry-2d
A regular octagon ABCDEFGH has sides of length two. Find the area of △ ADG.
AMC 10 2002B #18 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 10 2002B #19 grade 6+ arithmetic
Suppose that {a_n} is an arithmetic sequence with a₁+a₂+…+a₁₀₀=100 and a₁₀₁+a₁₀₂+…+a₂₀₀=200. What is the value of a₂ - a…
AMC 10 2002B #20 grade 8+ arithmetic
Let a, b, and c be real numbers such that a-7b+8c=4 and 8a+4b-c=7. Then a²-b²+c² is
AMC 10 2002B #21 grade 6+ rate-ratio
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 10 2002B #22 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 10 2002B #23 grade 6+ arithmetic
Let {a_k} be a sequence of integers such that a₁=1 and a_(m+n)=a_m+a_n+mn, for all positive integers m and n. Then a₁₂ i…
AMC 10 2002B #24 grade 8+ geometry-2d
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius 20 feet and revolves at t…
AMC 10 2002B #25 grade 8+ arithmetic
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…

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