Competition · AMC preparation · step 4 of 4

AMC 10 2019A: 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 2019A #1 grade 6+ arithmetic
What is the value of 2^(0^(1⁹))+((2⁰)¹)⁹?
AMC 10 2019A #2 grade 5+ arithmetic
What is the hundreds digit of (20!-15!)?
AMC 10 2019A #3 grade 4+ geometry-2d
Ana and Bonita were born on the same date in different years, n years apart. Last year Ana was 5 times as old as Bonita.…
AMC 10 2019A #4 grade 3+ arithmetic
A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. What is…
AMC 10 2019A #5 grade 6+ number-theory
What is the greatest number of consecutive integers whose sum is 45?
AMC 10 2019A #6 grade 4+ geometry-2d
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is…
AMC 10 2019A #7 grade 6+ geometry-2d
Two lines with slopes 1/2 and 2 intersect at (2,2). What is the area of the triangle enclosed by these two lines and the…
AMC 10 2019A #8 grade 8+ geometry-2d
The figure below shows line ell with a regular, infinite, recurring pattern of squares and line segments. How many of th…
AMC 10 2019A #9 grade 6+ number-theory
What is the greatest three-digit positive integer n for which the sum of the first n positive integers is not a divisor…
AMC 10 2019A #10 grade 6+ geometry-2d
A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 one-foot square tiles. A bug walks from one…
AMC 10 2019A #11 grade 6+ arithmetic
How many positive integer divisors of 201⁹ are perfect squares or perfect cubes (or both)?
AMC 10 2019A #12 grade 6+ arithmetic
Melanie computes the mean μ, the median M, and the modes of the 365 values that are the dates in the months of 2019. Thu…
AMC 10 2019A #13 grade 8+ geometry-2d
Let △ ABC be an isosceles triangle with BC = AC and ∠ ACB = 40°. Construct the circle with diameter BC, and let D and E…
AMC 10 2019A #14 grade 8+ counting
For a set of four distinct lines in a plane, there are exactly N distinct points that lie on two or more of the lines. W…
AMC 10 2019A #15 grade 8+ number-theory
A sequence of numbers is defined recursively by a₁ = 1, a₂ = 3/7, and a_n=(a_(n-2) · a_(n-1))/(2a_(n-2) - a_(n-1))for al…
AMC 10 2019A #16 grade 8+ geometry-2d
The figure below shows 13 circles of radius 1 within a larger circle. All the intersections occur at points of tangency.…
AMC 10 2019A #17 grade 7+ geometry-3d
A child builds towers using identically shaped cubes of different colors. How many different towers with a height 8 cube…
AMC 10 2019A #18 grade 8+ number-theory
For some positive integer k, the repeating base-k representation of the (base-ten) fraction 7/51 is 0.23_k = 0.232323...…
AMC 10 2019A #19 grade 8+ arithmetic
What is the least possible value of (x+1)(x+2)(x+3)(x+4)+2019where x is a real number?
AMC 10 2019A #20 grade 7+ probability
The numbers 1,2,…,9 are randomly placed into the 9 squares of a 3 × 3 grid. Each square gets one number, and each of the…
AMC 10 2019A #21 grade 8+ geometry-3d
A sphere with center O has radius 6. A triangle with sides of length 15, 15, and 24 is situated in space so that each of…
AMC 10 2019A #22 grade 7+ probability
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads,…
AMC 10 2019A #23 grade 6+ counting
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game…
AMC 10 2019A #24 grade 8+ algebra
Let p, q, and r be the distinct roots of the polynomial x³ - 22x² + 80x - 67. It is given that there exist real numbers…
AMC 10 2019A #25 grade 8+ number-theory
For how many integers n between 1 and 50, inclusive, is ((n²-1)!)/((n!)ⁿ) an integer? (Recall that 0! = 1.)

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