Competition · AMC preparation · step 4 of 4

AMC 10 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 10 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 10 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 10 2019B #3 grade 6+ arithmetic
In a high school with 500 students, 40% of the seniors play a musical instrument, while 30% of the non-seniors do not pl…
AMC 10 2019B #4 grade 6+ algebra
All lines with equation ax+by=c such that a,b,c form an arithmetic progression pass through a common point. What are the…
AMC 10 2019B #5 grade 8+ geometry-2d
Triangle ABC lies in the first quadrant. Points A, B, and C are reflected across the line y=x to points A', B', and C',…
AMC 10 2019B #6 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 10 2019B #7 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 10 2019B #8 grade 8+ geometry-2d
The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the…
AMC 10 2019B #9 grade 8+ arithmetic
The function f is defined by f(x) = ⌊|x|⌋ - |⌊ x ⌋|for all real numbers x, where ⌊ r ⌋ denotes the greatest integer less…
AMC 10 2019B #10 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 10 2019B #11 grade 6+ rate-ratio
Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue t…
AMC 10 2019B #12 grade 6+ arithmetic
What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than 2019?
AMC 10 2019B #13 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 10 2019B #14 grade 6+ arithmetic
The base-ten representation for 19! is 121,6T5,100,40M,832,H00, where T, M, and H denote digits that are not given. What…
AMC 10 2019B #15 grade 8+ geometry-2d
Right triangles T₁ and T₂, have areas of 1 and 2, respectively. A side of T₁ is congruent to a side of T₂, and a differe…
AMC 10 2019B #16 grade 8+ geometry-2d
In △ ABC with a right angle at C, point D lies in the interior of AB and point E lies in the interior of BC so that AC=C…
AMC 10 2019B #17 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 10 2019B #18 grade 8+ rate-ratio
Henry decides one morning to do a workout, and he walks 3/4 of the way from his home to his gym. The gym is 2 kilometers…
AMC 10 2019B #19 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 10 2019B #20 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 10 2019B #21 grade 7+ probability
Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she…
AMC 10 2019B #22 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 10 2019B #23 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 10 2019B #24 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 10 2019B #25 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,…

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