AMC 12A Problem and Solution

2017 AMC 12A

2017 AMC 12A

2017 AMC 12A problems and solutions. The test was held on February 7, 2017.

Problem 1

Pablo buys popsicles for his friends. The store sells single popsicles for each, 3-popsicle boxes for , and 5-popsicle boxes for . What is the greatest number of popsicles that Pablo can buy with ?

Solution

By the greedy algorithm, we can take two 5-popsicle boxes and one 3-popsicle box with To prove that this is optimal, consider an upper bound as follows: at the rate of per 5 popsicles, we can get

.

popsicles, which is less than 14. .

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?

Solution

Let

be our two numbers. Then

. Thus,

.

.

Problem3

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?

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2017 AMC 12A

Solution

Taking the contrapositive of the statement \didn't get an A, he didn't get all of them right\

.

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?

Solution

Let represent how far Jerry walked, and represent how far Sylvia walked. Since the field is a square, and Jerry walked two sides of it, while Silvia walked the diagonal, we can simply define the side of the square field to be one, and find the distances they walked. Since Jerry walked two sides, Since Silvia walked the diagonal, she walked the hypotenuse of a 45, 45, 90 triangle with leg length 1. Thus,

We can then

take

.

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?

Solution

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2017 AMC 12A

Let the group of people who all know each other be , and let the group of people who know no one be . Handshakes occur between each pair between each pair of members in . Thus, the answer is

such that

and

, and

Solution - Complementary Counting

The number of handshakes will be equivalent to the difference between the number of total

interactions and the number of hugs, which are and , respectively. Thus, the total

amount of handshakes is

Problem6

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?

Solution

The quadrilateral cannot be a straight line. Thus, the fourth side must be longer than

and shorter than

= 25. This means Joy can use the 19 . However, she has already used the rods of

possible integer rod lengths that fall into length cm and

cm so the answer is

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2017 AMC 12A

Problem7

Define a function on the positive integers recursively by

,

?

if is even, and

if is odd and

greater than . What is

Solution

This is a recursive function, which means the function is used to evaluate itself. To solve this, we must identify the base case, odd,

. We also know that when is

. Thus we know

, and we add each recursive cycle,

, which is answer

.

. Thus we know that

that n will always be odd in the recursion of which there are

of. Thus the answer is

Problem8

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

Solution

Let the length be . Then, we see that the region is just the union of the cylinder with central axis and radius and the two hemispheres connected to each face of the cylinder (also with radius ). Thus the volume is

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