Competition · AMC preparation · step 4 of 4

AMC 12 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 12 2016A #1 grade 6+ arithmetic
What is the value of (11!-10!)/(9!)?
AMC 12 2016A #2 grade 8+ algebra
For what value of x does 10^x· 100^2x=1000⁵?
AMC 12 2016A #3 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 12 2016A #4 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 12 2016A #5 grade 6+ number-theory
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for…
AMC 12 2016A #6 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 12 2016A #7 grade 9+ geometry-2d
Which of these describes the graph of x²(x+y+1)=y²(x+y+1) ?
AMC 12 2016A #8 grade 6+ geometry-2d
Find the area of the shaded region.
AMC 12 2016A #9 grade 9+ geometry-2d
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each si…
AMC 12 2016A #10 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 12 2016A #11 grade 8+ counting
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talen…
AMC 12 2016A #12 grade 10+ geometry-2d
In △ ABC, AB = 6, BC = 7, and CA = 8. Point D lies on BC, and AD bisects ∠ BAC. Point E lies on AC, and BE bisects ∠ ABC…
AMC 12 2016A #13 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 12 2016A #14 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 12 2016A #15 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 12 2016A #16 grade 11+ algebra
The graphs of y=log₃ x, y=log_x 3, y=log_(1/3) x, and y=log_x 1/3 are plotted on the same set of axes. How many points i…
AMC 12 2016A #17 grade 10+ geometry-2d
Let ABCD be a square. Let E, F, G and H be the centers, respectively, of equilateral triangles with bases AB, BC, CD, an…
AMC 12 2016A #18 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 12 2016A #19 grade 7+ probability
Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the pos…
AMC 12 2016A #20 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 12 2016A #21 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 12 2016A #22 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?
AMC 12 2016A #23 grade 10+ probability
Three numbers in the interval [0,1] are chosen independently and at random. What is the probability that the chosen numb…
AMC 12 2016A #24 grade 11+ algebra
There is a smallest positive real number a such that there exists a positive real number b such that all the roots of th…
AMC 12 2016A #25 grade 9+ number-theory
Let k be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bern…

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