Competition · AMC preparation · step 4 of 4

AMC 12 2019B: 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 2019B #1 grade 5+ rate-ratio
Alicia had two containers. The first was 5/6 full of water and the second was empty. She poured all the water from the f…
AMC 12 2019B #2 grade 4+ logic
Consider the statement, "If n is not prime, then n-2 is prime." Which of the following values of n is a counterexample t…
AMC 12 2019B #3 grade 8+ geometry-2d
Which one of the following rigid transformations (isometries) maps the line segment AB onto the line segment A'B' so tha…
AMC 12 2019B #4 grade 6+ algebra
There is a positive integer n such that (n+1)! + (n+2)! = n! · 440. What is the sum of the digits of n?
AMC 12 2019B #5 grade 6+ number-theory
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of…
AMC 12 2019B #6 grade 8+ geometry-2d
In a given plane, points A and B are 10 units apart. How many points C are there in the plane such that the perimeter of…
AMC 12 2019B #7 grade 6+ arithmetic
What is the sum of all real numbers x for which the median of the numbers 4,6,8,17, and x is equal to the mean of those…
AMC 12 2019B #8 grade 9+ algebra
Let f(x) = x²(1-x)². What is the value of the sum f (1/2019 )-f (2/2019 )+f (3/2019 )-f (4/2019 )+… + f (2017/2019 ) - f…
AMC 12 2019B #9 grade 11+ algebra
For how many integral values of x can a triangle of positive area be formed having side lengths log₂ x, log₄ x, 3?
AMC 12 2019B #10 grade 5+ counting
The figure below is a map showing 12 cities and 17 roads connecting certain pairs of cities. Paula wishes to travel alon…
AMC 12 2019B #11 grade 10+ geometry-3d
How many unordered pairs of edges of a given cube determine a plane?
AMC 12 2019B #12 grade 11+ geometry-2d
Right triangle ACD with right angle at C is constructed outwards on the hypotenuse AC of isosceles right triangle ABC wi…
AMC 12 2019B #13 grade 8+ probability
A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that…
AMC 12 2019B #14 grade 6+ number-theory
Let S be the set of all positive integer divisors of 100,000. How many numbers are the product of two distinct elements…
AMC 12 2019B #15 grade 8+ geometry-2d
As shown in the figure, line segment AD is trisected by points B and C so that AB=BC=CD=2. Three semicircles of radius 1…
AMC 12 2019B #16 grade 7+ probability
There are lily pads in a row numbered 0 to 11, in that order. There are predators on lily pads 3 and 6, and a morsel of…
AMC 12 2019B #17 grade 11+ algebra
How many nonzero complex numbers z have the property that 0, z, and z³, when represented by points in the complex plane,…
AMC 12 2019B #18 grade 10+ geometry-3d
Square pyramid ABCDE has base ABCD, which measures 3 cm on a side, and altitude AE perpendicular to the base, which meas…
AMC 12 2019B #19 grade 7+ probability
Raashan, Sylvia, and Ted play the following game. Each starts with $1. A bell rings every 15 seconds, at which time each…
AMC 12 2019B #20 grade 8+ geometry-2d
Points A=(6,13) and B=(12,11) lie on circle ω in the plane. Suppose that the tangent lines to ω at A and B intersect at…
AMC 12 2019B #21 grade 11+ algebra
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficient…
AMC 12 2019B #22 grade 8+ algebra
Define a sequence recursively by x₀=5 and x_(n+1)=(x_n²+5x_n+4)/(x_n+6) for all nonnegative integers n. Let m be the lea…
AMC 12 2019B #23 grade 7+ counting
How many sequences of 0s and 1s of length 19 are there that begin with a 0, end with a 0, contain no two consecutive 0s,…
AMC 12 2019B #24 grade 11+ geometry-2d
Let ω=-1/2+1/2i√3. Let S denote all points in the complex plane of the form a+bω+cω², where 0≤ a ≤ 1,0≤ b≤ 1, and 0≤ c≤…
AMC 12 2019B #25 grade 11+ geometry-2d
Let ABCD be a convex quadrilateral with BC=2 and CD=6. Suppose that the centroids of △ ABC,△ BCD, and △ ACD form the ver…

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