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AMC 10 2017 A

Question 1

What is the value of ?

Solution

  
  2020-07-09 06:36:08

Question 2

Pablo buys popsicles for his friends. The store sells single popsicles for each, -popsicle boxes for each, and -popsicle boxes for . What is the greatest number of popsicles that Pablo can buy with ?

Solution

  
  2020-07-09 06:36:08

Question 3

Tamara has three rows of two -feet by -feet flower beds in her garden. The beds are separated and also surrounded by -foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?


Solution

  
  2020-07-09 06:36:08

Question 4

Mia is ???helping??? her mom pick up toys that are strewn on the floor. Mia???s mom manages to put toys into the toy box every seconds, but each time immediately after those seconds have elapsed, Mia takes toys out of the box. How much time, in minutes, will it take Mia and her mom to put all toys into the box for the first time?

Solution

  
  2020-07-09 06:36:08

Question 5

The sum of two nonzero real numbers is times their product. What is the sum of the reciprocals of the two numbers?

Solution

  
  2020-07-09 06:36:08

Question 6

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically?

Solution

  
  2020-07-09 06:36:08

Question 7

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?

Solution

  
  2020-07-09 06:36:08

Question 8

At a gathering of people, there are people who all know each other and people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?

Solution

  
  2020-07-09 06:36:08

Question 9

Minnie rides on a flat road at kilometers per hour (kph), downhill at kph, and uphill at kph. Penny rides on a flat road at kph, downhill at kph, and uphill at kph. Minnie goes from town to town , a distance of km all uphill, then from town to town , a distance of km all downhill, and then back to town , a distance of km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the -km ride than it takes Penny?

Solution

  
  2020-07-09 06:36:08

Question 10

Joy has thin rods, one each of every integer length from cm through cm. She places the rods with lengths cm, cm, and cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?

Solution

  
  2020-07-09 06:36:08

Question 11

The region consisting of all points in three-dimensional space within units of line segment has volume . What is the length ?

Solution

  
  2020-07-09 06:36:08

Question 12

Let be a set of points in the coordinate plane such that two of the three quantities and are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for

Solution

  
  2020-07-09 06:36:08

Question 13

Define a sequence recursively by and the remainder when is divided by for all Thus the sequence starts What is

Solution

  
  2020-07-09 06:36:08

Question 14

Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was dollars. The cost of his movie ticket was of the difference between and the cost of his soda, while the cost of his soda was of the difference between and the cost of his movie ticket. To the nearest whole percent, what fraction of did Roger pay for his movie ticket and soda?

Solution

  
  2020-07-09 06:36:08

Question 15

Chlo?? chooses a real number uniformly at random from the interval . Independently, Laurent chooses a real number uniformly at random from the interval . What is the probability that Laurent's number is greater than Chlo??'s number? (Assume they cannot be equal)

Solution

  
  2020-07-09 06:36:08

Question 16

There are 10 horses, named Horse 1, Horse 2, , Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse runs one lap in exactly minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time , in minutes, at which all 10 horses will again simultaneously be at the starting point is . Let be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits of ?

Solution

  
  2020-07-09 06:36:08

Question 17

Distinct points , , , lie on the circle and have integer coordinates. The distances and are irrational numbers. What is the greatest possible value of the ratio ?

Solution

  
  2020-07-09 06:36:08

Question 18

Amelia has a coin that lands heads with probability , and Blaine has a coin that lands on heads with probability . Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is , where and are relatively prime positive integers. What is ?

Solution

  
  2020-07-09 06:36:08

Question 19

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?

Solution

  
  2020-07-09 06:36:08

Question 20

Let equal the sum of the digits of positive integer . For example, . For a particular positive integer , . Which of the following could be the value of ?

Solution

  
  2020-07-09 06:36:08

Question 21


A square with side length is inscribed in a right triangle with sides of length , , and so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length is inscribed in another right triangle with sides of length , , and so that one side of the square lies on the hypotenuse of the triangle. What is ?

Solution

  
  2020-07-09 06:36:08

Question 22

Sides and of equilateral triangle are tangent to a circle at points and respectively. What fraction of the area of lies outside the circle?

Solution

  
  2020-07-09 06:36:08

Question 23


How many triangles with positive area have all their vertices at points in the coordinate plane, where and are integers between and , inclusive?

Solution

  
  2020-07-09 06:36:08

Question 24


For certain real numbers , , and , the polynomial has three distinct roots, and each root of is also a root of the polynomial What is ?

Solution

  
  2020-07-09 06:36:08

Question 25

How many integers between and , inclusive, have the property that some permutation of its digits is a multiple of between and For example, both and have this property.

Solution

  
  2020-07-09 06:36:08

Answer Keys


Question 1: C
Question 2: D
Question 3: B
Question 4: B
Question 5: C
Question 6: B
Question 7: A
Question 8: B
Question 9: C
Question 10: B
Question 11: D
Question 12: E
Question 13: D
Question 14: D
Question 15: C
Question 16: B
Question 17: D
Question 18: D
Question 19: C
Question 20: D
Question 21: D
Question 22: E
Question 23: B
Question 24: C
Question 25: A