Competition · AMC preparation · step 4 of 4

AMC 10 2017A: 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 2017A #1 grade 5+ arithmetic
What is the value of (2(2(2(2(2(2+1)+1)+1)+1)+1)+1)?
AMC 10 2017A #2 grade 4+ arithmetic
Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2 each, and 5-…
AMC 10 2017A #3 grade 4+ geometry-2d
Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1…
AMC 10 2017A #4 grade 5+ rate-ratio
Mia is "helping" her mom pick up 30 toys that are strewn on the floor. Mia’s mom manages to put 3 toys into the toy box…
AMC 10 2017A #5 grade 7+ algebra
The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?
AMC 10 2017A #6 grade 6+ number-theory
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A…
AMC 10 2017A #7 grade 8+ geometry-2d
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east…
AMC 10 2017A #8 grade 4+ counting
At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know…
AMC 10 2017A #9 grade 7+ rate-ratio
Minnie rides on a flat road at 20 kilometers per hour (kph), downhill at 30 kph, and uphill at 5 kph. Penny rides on a f…
AMC 10 2017A #10 grade 7+ geometry-2d
Joy has 30 thin rods, one each of every integer length from 1 cm through 30 cm. She places the rods with lengths 3 cm, 7…
AMC 10 2017A #11 grade 8+ geometry-3d
The region consisting of all points in three-dimensional space within 3 units of line segment AB has volume 216π. What i…
AMC 10 2017A #12 grade 8+ geometry-2d
Let S be a set of points (x,y) in the coordinate plane such that two of the three quantities 3, x+2, and y-4 are equal a…
AMC 10 2017A #13 grade 4+ number-theory
Define a sequence recursively by F₀=0, F₁=1, and F_n= the remainder when F_(n-1)+F_(n-2) is divided by 3, for all n≥ 2.…
AMC 10 2017A #14 grade 8+ rate-ratio
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was A dollars. Th…
AMC 10 2017A #15 grade 7+ probability
Chloe chooses a real number uniformly at random from the interval [0, 2017]. Independently, Laurent chooses a real numbe…
AMC 10 2017A #16 grade 4+ number-theory
There are 10 horses, named Horse 1, Horse 2, . . . , Horse 10. They get their names from how many minutes it takes them…
AMC 10 2017A #17 grade 8+ geometry-2d
Distinct points P, Q, R, S lie on the circle x²+y²=25 and have integer coordinates. The distances PQ and RS are irration…
AMC 10 2017A #18 grade 8+ probability
Amelia has a coin that lands heads with probability 1/3, and Blaine has a coin that lands on heads with probability 2/5.…
AMC 10 2017A #19 grade 4+ counting
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the fiv…
AMC 10 2017A #20 grade 6+ number-theory
Let S(n) equal the sum of the digits of positive integer n. For example, S(1507) = 13. For a particular positive integer…
AMC 10 2017A #21 grade 8+ geometry-2d
A square with side length x is inscribed in a right triangle with sides of length 3, 4, and 5 so that one vertex of the…
AMC 10 2017A #22 grade 8+ geometry-2d
Sides AB and AC of equilateral triangle ABC are tangent to a circle at points B and C respectively. What fraction of the…
AMC 10 2017A #23 grade 8+ geometry-2d
How many triangles with positive area have all their vertices at points (i,j) in the coordinate plane, where i and j are…
AMC 10 2017A #24 grade 8+ algebra
For certain real numbers a, b, and c, the polynomial g(x) = x³ + ax² + x + 10has three distinct roots, and each root of…
AMC 10 2017A #25 grade 4+ counting
How many integers between 100 and 999, inclusive, have the property that some permutation of its digits is a multiple of…

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