Competition · AMC preparation · step 4 of 4

AMC 10 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 10 2011B #1 grade 5+ arithmetic
What is (2+4+6)/(1+3+5) - (1+3+5)/(2+4+6) ?
AMC 10 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 10 2011B #3 grade 5+ geometry-2d
At a store, when a length is reported as x inches that means the length is at least x - 0.5 inches and at most x + 0.5 i…
AMC 10 2011B #4 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 10 2011B #5 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 10 2011B #6 grade 5+ arithmetic
On Halloween Casper ate 1/3 of his candies and then gave 2 candies to his brother. The next day he ate 1/3 of his remain…
AMC 10 2011B #7 grade 8+ geometry-2d
The sum of two angles of a triangle is 6/5 of a right angle, and one of these two angles is 30° larger than the other. W…
AMC 10 2011B #8 grade 6+ arithmetic
At a certain beach if it is at least 80° F and sunny, then the beach will be crowded. On June 10 the beach was not crowd…
AMC 10 2011B #9 grade 8+ geometry-2d
The area of triangleEBD is one third of the area of △ABC. Segment DE is perpendicular to segment AB. What is BD?
AMC 10 2011B #10 grade 6+ rate-ratio
Consider the set of numbers {1, 10, 10², 10³, …, 10¹⁰}. The ratio of the largest element of the set to the sum of the ot…
AMC 10 2011B #11 grade 6+ arithmetic
There are 52 people in a room. what is the largest value of n such that the statement "At least n people in this room ha…
AMC 10 2011B #12 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 10 2011B #13 grade 7+ probability
Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the prod…
AMC 10 2011B #14 grade 8+ geometry-2d
A rectangular parking lot has a diagonal of 25 meters and an area of 168 square meters. In meters, what is the perimeter…
AMC 10 2011B #15 grade 6+ arithmetic
Let @ denote the "averaged with" operation: a @ b = (a+b)/2. Which of the following distributive laws hold for all numbe…
AMC 10 2011B #16 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 10 2011B #17 grade 8+ geometry-2d
In the given circle, the diameter EB is parallel to DC, and AB is parallel to ED. The angles AEB and ABE are in the rati…
AMC 10 2011B #18 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 10 2011B #19 grade 8+ algebra
What is the product of all the roots of the equation √(5 | x | + 8) = √(x² - 16).
AMC 10 2011B #20 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 10 2011B #21 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 10 2011B #22 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 10 2011B #23 grade 8+ arithmetic
What is the hundreds digit of 2011²⁰¹¹?
AMC 10 2011B #24 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 10 2011B #25 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…

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