Competition · AMC preparation · step 4 of 4

AMC 12 2011B: 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 2011B #1 grade 5+ arithmetic
What is (2+4+6)/(1+3+5) - (1+3+5)/(2+4+6) ?
AMC 12 2011B #2 grade 6+ arithmetic
Josanna's test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise here test average at least 3 points with…
AMC 12 2011B #3 grade 7+ arithmetic
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them…
AMC 12 2011B #4 grade 4+ number-theory
In multiplying two positive integers a and b, Ron reversed the digits of the two-digit number a. His erroneous product w…
AMC 12 2011B #5 grade 6+ number-theory
Let N be the second smallest positive integer that is divisible by every positive integer less than 7. What is the sum o…
AMC 12 2011B #6 grade 10+ geometry-2d
Two tangents to a circle are drawn from a point A. The points of contact B and C divide the circle into arcs with length…
AMC 12 2011B #7 grade 7+ rate-ratio
Let x and y be two-digit positive integers with mean 60. What is the maximum value of the ratio x/y?
AMC 12 2011B #8 grade 7+ rate-ratio, geometry-2d
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and t…
AMC 12 2011B #9 grade 7+ probability
Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the prod…
AMC 12 2011B #10 grade 8+ geometry-2d
Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so that ∠ AMD = ∠ CMD. What is the degree measure of…
AMC 12 2011B #11 grade 8+ number-theory, geometry-2d
A frog located at (x,y), with both x and y integers, makes successive jumps of length 5 and always lands on points with…
AMC 12 2011B #12 grade 8+ geometry-2d, probability
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally like…
AMC 12 2011B #13 grade 7+ logic, algebra
Brian writes down four integers w > x > y > z whose sum is 44. The pairwise positive differences of these numbers are 1,…
AMC 12 2011B #14 grade 11+ geometry-2d
A segment through the focus F of a parabola with vertex V is perpendicular to FV and intersects the parabola in points A…
AMC 12 2011B #15 grade 8+ number-theory, counting
How many positive two-digit integers are factors of 2²⁴-1? ~ pi_is_3.14
AMC 12 2011B #16 grade 8+ geometry-2d
Rhombus ABCD has side length 2 and ∠ B = 120°. Region R consists of all points inside the rhombus that are closer to ver…
AMC 12 2011B #17 grade 11+ algebra, pattern
Let f(x) = 10^10x, g(x) = log₁₀(x/10), h₁(x) = g(f(x)), and h_n(x) = h₁(h_(n-1)(x)) for integers n ≥ 2. What is the sum…
AMC 12 2011B #18 grade 8+ geometry-3d
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is place…
AMC 12 2011B #19 grade 8+ number-theory, geometry-2d
A lattice point in an xy-coordinate system is any point (x, y) where both x and y are integers. The graph of y = mx +2 p…
AMC 12 2011B #20 grade 10+ geometry-2d
Triangle ABC has AB = 13, BC = 14, and AC = 15. The points D, E, and F are the midpoints of AB, BC, and AC respectively.…
AMC 12 2011B #21 grade 9+ number-theory, algebra
The arithmetic mean of two distinct positive integers x and y is a two-digit integer. The geometric mean of x and y is o…
AMC 12 2011B #22 grade 8+ geometry-2d, pattern
Let T₁ be a triangle with side lengths 2011, 2012, and 2013. For n ≥ 1, if T_n = △ ABC and D, E, and F are the points of…
AMC 12 2011B #23 grade 7+ counting, geometry-2d
A bug travels in the coordinate plane, moving only along the lines that are parallel to the x-axis or y-axis. Let A = (-…
AMC 12 2011B #24 grade 11+ algebra, geometry-2d
Let P(z) = z⁸ + (4√3 + 6)z⁴ - (4√3 + 7). What is the minimum perimeter among all the 8-sided polygons in the complex pla…
AMC 12 2011B #25 grade 7+ probability, number-theory
For every m and k integers with k odd, denote by [m/k] the integer closest to m/k. For every odd integer k, let P(k) be…

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