Competition · AMC preparation · step 4 of 4

AMC 10 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 10 2016B #1 grade 8+ algebra
What is the value of (2a⁻¹+(a⁻¹)/2)/a when a= 1/2?
AMC 10 2016B #2 grade 6+ arithmetic
If n♡ m=n³m², what is (2♡ 4)/(4♡ 2)?
AMC 10 2016B #3 grade 7+ arithmetic
Let x=-2016. What is the value of || |x|-x|-|x||-x ?
AMC 10 2016B #4 grade 4+ rate-ratio
Zoey read 15 books, one at a time. The first book took her 1 day to read, the second book took her 2 days to read, the t…
AMC 10 2016B #5 grade 6+ arithmetic
The mean age of Amanda's 4 cousins is 8, and their median age is 5. What is the sum of the ages of Amanda's youngest and…
AMC 10 2016B #6 grade 4+ arithmetic
Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-dig…
AMC 10 2016B #7 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 10 2016B #8 grade 5+ arithmetic
What is the tens digit of 2015²⁰¹⁶-2017?
AMC 10 2016B #9 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 10 2016B #10 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 10 2016B #11 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 10 2016B #12 grade 7+ probability
Two different numbers are selected at random from {1, 2, 3, 4, 5} and multiplied together. What is the probability that…
AMC 10 2016B #13 grade 6+ number-theory
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets acco…
AMC 10 2016B #14 grade 8+ counting
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely…
AMC 10 2016B #15 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 10 2016B #16 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 10 2016B #17 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 10 2016B #18 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 10 2016B #19 grade 8+ geometry-2d
Rectangle ABCD has AB=5 and BC=4. Point E lies on AB so that EB=1, point G lies on BC so that CG=1, and point F lies on…
AMC 10 2016B #20 grade 8+ geometry-2d
A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius 2 centere…
AMC 10 2016B #21 grade 8+ geometry-2d
What is the area of the region enclosed by the graph of the equation x²+y²=|x|+|y|?
AMC 10 2016B #22 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 10 2016B #23 grade 8+ geometry-2d
In regular hexagon ABCDEF, points W, X, Y, and Z are chosen on sides BC, CD, EF, and FA respectively, so lines AB, ZW, Y…
AMC 10 2016B #24 grade 7+ arithmetic
How many four-digit integers abcd, with a ≠ 0, have the property that the three two-digit integers ab<bc<cd form an incr…
AMC 10 2016B #25 grade 8+ arithmetic
Let f(x)=sum_(k=2)¹⁰(⌊ kx ⌋ -k ⌊ x ⌋), where ⌊ r ⌋ denotes the greatest integer less than or equal to r. How many distin…

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