Competition · AMC preparation · step 4 of 4

AMC 10 2002A: 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 2002A #1 grade 8+ rate-ratio
The ratio (10²⁰⁰⁰+10²⁰⁰²)/(10²⁰⁰¹+10²⁰⁰¹) is closest to which of the following numbers?
AMC 10 2002A #2 grade 6+ arithmetic
Given that a, b, and c are non-zero real numbers, define (a, b, c) = a/b + b/c + c/a, find (2, 12, 9).
AMC 10 2002A #3 grade 6+ arithmetic
According to the standard convention for exponentiation, 2^(2^2²) = 2^(2^(2²)) = 2¹⁶ = 65536. If the order in which the…
AMC 10 2002A #4 grade 6+ arithmetic
For how many positive integers m does there exist at least one positive integer n such that m · n ≤ m + n?
AMC 10 2002A #5 grade 7+ geometry-2d
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround…
AMC 10 2002A #6 grade 5+ arithmetic
Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtrac…
AMC 10 2002A #7 grade 7+ geometry-2d
A 45° arc of circle A is equal in length to a 30° arc of circle B. What is the ratio of circle A's area and circle B's a…
AMC 10 2002A #8 grade 6+ geometry-2d
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let B be the total a…
AMC 10 2002A #9 grade 7+ arithmetic
There are 3 numbers A, B, and C, such that 1001C - 2002A = 4004, and 1001B + 3003A = 5005. What is the average of A, B,…
AMC 10 2002A #10 grade 8+ arithmetic
Compute the sum of all the roots of (2x+3)(x-4)+(2x+3)(x-6)=0
AMC 10 2002A #11 grade 5+ arithmetic
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up…
AMC 10 2002A #12 grade 8+ rate-ratio
Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he w…
AMC 10 2002A #13 grade 8+ geometry-2d
Given a triangle with side lengths 15, 20, and 25, find the triangle's shortest altitude.
AMC 10 2002A #14 grade 6+ number-theory
Both roots of the quadratic equation x² - 63x + k = 0 are prime numbers. The number of possible values of k is
AMC 10 2002A #15 grade 4+ number-theory
Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum…
AMC 10 2002A #16 grade 8+ arithmetic
Let a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5. What is a + b + c + d?
AMC 10 2002A #17 grade 6+ rate-ratio
Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size.…
AMC 10 2002A #18 grade 6+ geometry-3d
A 3x3x3 cube is made of 27 normal dice. Each die's opposite sides sum to 7. What is the smallest possible sum of all of…
AMC 10 2002A #19 grade 7+ geometry-2d
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-…
AMC 10 2002A #20 grade 8+ geometry-2d
Points A,B,C,D,E and F lie, in that order, on AF, dividing it into five segments, each of length 1. Point G is not on li…
AMC 10 2002A #21 grade 6+ arithmetic
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that…
AMC 10 2002A #22 grade 8+ number-theory
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with…
AMC 10 2002A #23 grade 8+ geometry-2d
Points A,B,C and D lie on a line, in that order, with AB = CD and BC = 12. Point E is not on the line, and BE = CE = 10.…
AMC 10 2002A #24 grade 7+ probability
Tina randomly selects two distinct numbers from the set { 1, 2, 3, 4, 5 }, and Sergio randomly selects a number from the…
AMC 10 2002A #25 grade 8+ geometry-2d
In trapezoid ABCD with bases AB and CD, we have AB = 52, BC = 12, CD = 39, and DA = 5. The area of ABCD is

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