Competition · AMC preparation · step 4 of 4

AMC 12 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 12 2002A #1 grade 8+ arithmetic
Compute the sum of all the roots of (2x+3)(x-4)+(2x+3)(x-6)=0
AMC 12 2002A #2 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 12 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 12 2002A #4 grade 7+ geometry-2d
Find the degree measure of an angle whose complement is 25% of its supplement.
AMC 12 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 12 2002A #6 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 12 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 12 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 12 2002A #9 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 12 2002A #10 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 12 2002A #11 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 12 2002A #12 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 12 2002A #13 grade 9+ algebra
Two different positive numbers a and b each differ from their reciprocals by 1. What is a+b?
AMC 12 2002A #14 grade 11+ algebra
For all positive integers n, let f(n)=log₂₀₀₂ n². Let N=f(11)+f(13)+f(14). Which of the following relations is true?
AMC 12 2002A #15 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 12 2002A #16 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 12 2002A #17 grade 5+ number-theory
Several sets of prime numbers, such as {7,83,421,659} use each of the nine nonzero digits exactly once. What is the smal…
AMC 12 2002A #18 grade 10+ geometry-2d
Let C₁ and C₂ be circles defined by (x-10)² + y² = 36 and (x+15)² + y² = 81 respectively. What is the length of the shor…
AMC 12 2002A #19 grade 9+ algebra
The graph of the function f is shown below. How many solutions does the equation f(f(x))=6 have?
AMC 12 2002A #20 grade 8+ number-theory
Suppose that a and b are digits, not both nine and not both zero, and the repeating decimal 0.ab is expressed as a fract…
AMC 12 2002A #21 grade 6+ pattern
Consider the sequence of numbers: 4,7,1,8,9,7,6,… For n>2, the n-th term of the sequence is the units digit of the sum o…
AMC 12 2002A #22 grade 10+ geometry-2d
Triangle ABC is a right triangle with ∠ ACB as its right angle, m∠ ABC = 60° , and AB = 10. Let P be randomly chosen ins…
AMC 12 2002A #23 grade 10+ geometry-2d
In triangle ABC, side AC and the perpendicular bisector of BC meet in point D, and BD bisects ∠ ABC. If AD=9 and DC=7, w…
AMC 12 2002A #24 grade 12+ algebra
Find the number of ordered pairs of real numbers (a,b) such that (a+bi)²⁰⁰² = a-bi.
AMC 12 2002A #25 grade 9+ algebra
The nonzero coefficients of a polynomial P with real coefficients are all replaced by their mean to form a polynomial Q.…

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