Competition · AMC preparation · step 4 of 4

AMC 12 2016B: 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 2016B #1 grade 8+ algebra
What is the value of (2a⁻¹+(a⁻¹)/2)/a when a= 1/2?
AMC 12 2016B #2 grade 6+ arithmetic
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of 1 a…
AMC 12 2016B #3 grade 7+ arithmetic
Let x=-2016. What is the value of || |x|-x|-|x||-x ?
AMC 12 2016B #4 grade 8+ rate-ratio
The ratio of the measures of two acute angles is 5:4, and the complement of one of these two angles is twice as large as…
AMC 12 2016B #5 grade 4+ arithmetic
The War of 1812 started with a declaration of war on Thursday, June 18, 1812. The peace treaty to end the war was signed…
AMC 12 2016B #6 grade 8+ geometry-2d
All three vertices of bigtriangleup ABC lie on the parabola defined by y=x², with A at the origin and BC parallel to the…
AMC 12 2016B #7 grade 6+ counting
Josh writes the numbers 1,2,3,…,99,100. He marks out 1, skips the next number (2), marks out 3, and continues skipping a…
AMC 12 2016B #8 grade 7+ geometry-2d
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length 3 inches weighs 12 ounc…
AMC 12 2016B #9 grade 6+ geometry-2d
Carl decided to fence in his rectangular garden. He bought 20 fence posts, placed one on each of the four corners, and s…
AMC 12 2016B #10 grade 8+ geometry-2d
A quadrilateral has vertices P(a,b), Q(b,a), R(-a, -b), and S(-b, -a), where a and b are integers with a>b>0. The area o…
AMC 12 2016B #11 grade 8+ counting
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely…
AMC 12 2016B #12 grade 4+ geometry-2d
All the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are written in a 3×3 array of squares, one number in each square, in such a wa…
AMC 12 2016B #13 grade 10+ geometry-2d
Alice and Bob live 10 miles apart. One day Alice looks due north from her house and sees an airplane. At the same time B…
AMC 12 2016B #14 grade 8+ algebra
The sum of an infinite geometric series is a positive number S, and the second term in the series is 1. What is the smal…
AMC 12 2016B #15 grade 6+ geometry-3d
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight…
AMC 12 2016B #16 grade 6+ counting
In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers?
AMC 12 2016B #17 grade 10+ geometry-2d
In △ ABC shown in the figure, AB=7, BC=8, CA=9, and AH is an altitude. Points D and E lie on sides AC and AB, respective…
AMC 12 2016B #18 grade 8+ geometry-2d
What is the area of the region enclosed by the graph of the equation x²+y²=|x|+|y|?
AMC 12 2016B #19 grade 8+ probability
Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he g…
AMC 12 2016B #20 grade 7+ arithmetic
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won 10…
AMC 12 2016B #21 grade 10+ geometry-2d
Let ABCD be a unit square. Let Q₁ be the midpoint of CD. For i=1,2,…, let P_i be the intersection of AQ_i and BD, and le…
AMC 12 2016B #22 grade 7+ number-theory
For a certain positive integer n less than 1000, the decimal equivalent of 1/n is 0.abcdef, a repeating decimal of perio…
AMC 12 2016B #23 grade 10+ geometry-3d
What is the volume of the region in three-dimensional space defined by the inequalities |x|+|y|+|z|≤1 and |x|+|y|+|z-1|≤…
AMC 12 2016B #24 grade 9+ number-theory
There are exactly 77,000 ordered quadruplets (a, b, c, d) such that gcd(a, b, c, d) = 77 and lcm(a, b, c, d) = n. What i…
AMC 12 2016B #25 grade 11+ algebra
The sequence (a_n) is defined recursively by a₀=1, a₁=¹⁹√2, and a_n=a_(n-1)a_(n-2)² for n≥ 2. What is the smallest posit…

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