2017 AMC 12A Problems
NOTE: AS OF NOW A WORK IN PROGRESS (Problems are not accurate/might not be formatted correctly)
2017 AMC 12A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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Contents
- 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
- 23 Problem 23
Problem 1
Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2, and 5-popsicle boxes for $3. What is the greatest number of popsicles that Pablo can buy with $8?
Problem 2
The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?
Problem 3
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 one of these statements necessarily follows logically?
Problem 4
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?
Problem 5
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?
Problem 6
Joy has thin rods, one each of every integer length from through . She places the rods with lengths , , and 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?
Problem 7
Define a function on the positive integers recursively by , if is even, and if is odd and greater than . What is ?
Problem 8
The region consisting of all points in three-dimensional space within units of line segment has volume . What is the length ?
Problem 9
Let be the 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 the common value. Which of the following is a correct description of ?
Problem 10
Chloé 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 Chloe's number?
Problem 11
Each of the students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are students who cannot sing, students who cannot dance, and students who cannot act. How many students have two of these talents?
Problem 12
In , , , and . Point lies on , and bisects . Point lies on , and bisects . The bisectors intersect at . What is the ratio : ?
Problem 13
Let be a positive multiple of . One red ball and green balls are arranged in a line in random order. Let be the probability that at least of the green balls are on the same side of the red ball. Observe that and that approaches as grows large. What is the sum of the digits of the least value of such that ?
Problem 14
Each vertex of a cube is to be labeled with an integer from through , with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
Problem 15
Circles with centers and , having radii and , respectively, lie on the same side of line and are tangent to at and , respectively, with between and . The circle with center is externally tangent to each of the other two circles. What is the area of triangle ?
Problem 16
The graphs of and are plotted on the same set of axes. How many points in the plane with positive -coordinates lie on two or more of the graphs?
Problem 17
Let be a square. Let and be the centers, respectively, of equilateral triangles with bases and each exterior to the square. What is the ratio of the area of square to the area of square ?
Problem 18
For some positive integer the number has positive integer divisors, including and the number How many positive integer divisors does the number have?
Problem 19
Jerry starts at on the real number line. He tosses a fair coin times. When he gets heads, he moves unit in the positive direction; when he gets tails, he moves unit in the negative direction. The probability that he reaches at some time during this process is where and are relatively prime positive integers. What is (For example, he succeeds if his sequence of tosses is )
Problem 20
A binary operation has the properties that and that for all nonzero real numbers and (Here the dot represents the usual multiplication operation.) The solution to the equation can be written as where and are relatively prime positive integers. What is
Problem 21
A quadrilateral is inscribed in a circle of radius Three of the sides of this quadrilateral have length What is the length of its fourth side?
Problem 22
How many ordered triples of positive integers satisfy and ?
Problem 23
Three numbers in the interval are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.