1、2017 AMC 10A Problems 2017 AMC 10A (Answer Key) Printable version: Wiki | AoPS Resources PDF Instructions 1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 2. You will receive 6 points for each correct answer,
2、2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. 3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculato
3、rs that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator). 4. Figures are not necessarily drawn to scale. 5. You will have 75 minutes working time to complete the test. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Conte
4、nts hide 1 Problem 1 2 Problem 2 3 Problem 3 4 Problem 4 5 Problem 5 6 Problem 6 7 Problem 7 8 Problem 8 9 Problem 9 10 Problem 10 11 Problem 11 12 Problem 12 13 Problem 13 14 Problem 14 15 Problem 15 16 Problem 16 17 Problem 17 18 Problem 18 19 Problem 19 20 Problem 20 21 Problem 21 22 Problem 22 2
5、3 Problem 23 24 Problem 24 25 Problem 25 26 See also Problem 1 What is the value of ? Solution Problem 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 wi
6、th ? Solution Problem 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 Problem 4 Mia is “helping” her mom pick up toys
7、that are strewn on the floor. Mias 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 Prob
8、lem 5 The sum of two nonzero real numbers is times their product. What is the sum of the reciprocals of the two numbers? Solution Problem 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 of these statemen
9、ts necessarily follows logically? Solution Problem 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 th
10、e following is closest to how much shorter Silvias trip was, compared to Jerrys trip? Solution Problem 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 handshak
11、es occur? Solution Problem 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
12、 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 Problem 10 Joy has thin rods, one each of every integer length from cm through cm. S
13、he 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 Problem 11 The region consisting of all points in t
14、hree-dimensional space within units of line segment has volume . What is the length ? Solution Problem 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 follow
15、ing is a correct description for Solution Problem 13 Define a sequence recursively by and the remainder when is divided by for all Thus the sequence starts What is Solution Problem 14 Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Rogers allowance was dollars. T
16、he 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 Problem 15 Chlo chooses
17、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 Laurents number is greater than Chlos number? Solution Problem 16 There are 10 horses, named Horse 1, Horse 2, , Horse 10. They get
18、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 trac
19、k 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 Problem 17 Distin
20、ct 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 Problem 18 Amelia has a coin that lands heads with probability , and Blaine has a coin that lands on heads with probability . Amelia a
21、nd 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 Problem 19 Alice refuses to sit next to e
22、ither 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 Problem 20 Let equal the sum of the digits of positive integer . For example, . For a particular positive integer , . Which of the followin
23、g could be the value of ? Solution Problem 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
24、 length , , and so that one side of the square lies on the hypotenuse of the triangle. What is ? Solution Problem 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 Problem 23 How many triangles with
25、 positive area have all their vertices at points in the coordinate plane, where and are integers between and , inclusive? Solution Problem 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 Problem 25 Ho
26、w 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 2017 AMC 10A Answer Key 1. C 2. D 3. B 4. B 5. C 6. B 7. A 8. B 9. C 10. B 11. D 12. E 13. D 14. D 15. C 16. B 17. D 18. D 19
27、. C 20. D 21. D 22. E 23. B 24. C 25. A ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Search 2017 AMC 10A Problems/Problem 1 Contents hide 1 Problem 2 Solution 1 3 Solution 2 4 Solution 3 5 Solution 4 6 See Also Problem What is the value
28、 of ? Solution 1 Notice this is the term in a recursive sequence, defined recursively as Thus: Solution 2 Starting to compute the inner expressions, we see the results are . This is always less than a power of . The only admissible answer choice by this rule is thus . Solution 3 Working our way from the innermost parenthesis outwards and directly computing, we have . Solution 4
