Competition · AMC preparation · step 4 of 4

AMC 10 2016A: 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 2016A #1 grade 6+ arithmetic
What is the value of (11!-10!)/(9!)?
AMC 10 2016A #2 grade 8+ algebra
For what value of x does 10^x· 100^2x=1000⁵?
AMC 10 2016A #3 grade 7+ arithmetic
For every dollar Ben spent on bagels, David spent 25 cents less. Ben paid $12.50 more than David. How much did they spen…
AMC 10 2016A #4 grade 7+ number-theory
The remainder can be defined for all real numbers x and y with y ≠ 0 by rem (x ,y)=x-y ⌊ x/y ⌋where ⌊ x/y ⌋ denotes the…
AMC 10 2016A #5 grade 6+ geometry-3d
A rectangular box has integer side lengths in the ratio 1: 3: 4. Which of the following could be the volume of the box?
AMC 10 2016A #6 grade 4+ counting
Ximena lists the whole numbers 1 through 30 once. Emilio copies Ximena's numbers, replacing each occurrence of the digit…
AMC 10 2016A #7 grade 6+ arithmetic
The mean, median, and mode of the 7 data values 60, 100, x, 40, 50, 200, 90 are all equal to x. What is the value of x?
AMC 10 2016A #8 grade 6+ rate-ratio
Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as l…
AMC 10 2016A #9 grade 6+ algebra
A triangular array of 2016 coins has 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and s…
AMC 10 2016A #10 grade 6+ geometry-2d
A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmeti…
AMC 10 2016A #11 grade 6+ geometry-2d
Find the area of the shaded region.
AMC 10 2016A #12 grade 7+ probability
Three distinct integers are selected at random between 1 and 2016, inclusive. Which of the following is a correct statem…
AMC 10 2016A #13 grade 7+ counting
Five friends sat in a movie theater in a row containing 5 seats, numbered 1 to 5 from left to right. (The directions "le…
AMC 10 2016A #14 grade 6+ counting
How many ways are there to write 2016 as the sum of twos and threes, ignoring order? (For example, 1008· 2 + 0· 3 and 40…
AMC 10 2016A #15 grade 8+ geometry-2d
Seven cookies of radius 1 inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all…
AMC 10 2016A #16 grade 8+ geometry-2d
A triangle with vertices A(0, 2), B(-3, 2), and C(-3, 0) is reflected about the x-axis, then the image △ A'B'C' is rotat…
AMC 10 2016A #17 grade 7+ probability
Let N be a positive multiple of 5. One red ball and N green balls are arranged in a line in random order. Let P(N) be th…
AMC 10 2016A #18 grade 7+ geometry-3d
Each vertex of a cube is to be labeled with an integer 1 through 8, with each integer being used once, in such a way tha…
AMC 10 2016A #19 grade 8+ geometry-2d
In rectangle ABCD, AB=6 and BC=3. Point E between B and C, and point F between E and C are such that BE=EF=FC. Segments…
AMC 10 2016A #20 grade 7+ algebra
For some particular value of N, when (a+b+c+d+1)^N is expanded and like terms are combined, the resulting expression con…
AMC 10 2016A #21 grade 8+ geometry-2d
Circles with centers P, Q and R, having radii 1, 2 and 3, respectively, lie on the same side of line l and are tangent t…
AMC 10 2016A #22 grade 6+ number-theory
For some positive integer n, the number 110n³ has 110 positive integer divisors, including 1 and the number 110n³. How m…
AMC 10 2016A #23 grade 8+ algebra
A binary operation ♢ has the properties that a ♢ (b ♢ c) = (a ♢ b)· c and that a ♢ a=1 for all nonzero real numbers a, b…
AMC 10 2016A #24 grade 8+ geometry-2d
A quadrilateral is inscribed in a circle of radius 200√2. Three of the sides of this quadrilateral have length 200. What…
AMC 10 2016A #25 grade 7+ number-theory
How many ordered triples (x,y,z) of positive integers satisfy lcm(x,y) = 72, lcm(x,z) = 600 and lcm(y,z)=900?

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