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AMC 10 2015 B

Question 1

What is the value of ?

Solution

  
  2020-07-09 06:35:43

Question 2

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?

Solution

  
  2020-07-09 06:35:43

Question 3

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is , and one of the numbers is . What is the other number?

Solution

  
  2020-07-09 06:35:43

Question 4

Four siblings ordered an extra large pizza. Alex ate , Beth , and Cyril of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?

Solution

  
  2020-07-09 06:35:43

Question 5

David, Hikmet, Jack, Marta, Rand, and Todd were in a -person race with other people. Rand finished places ahead of Hikmet. Marta finished place behind Jack. David finished places behind Hikmet. Jack finished places behind Todd. Todd finished place behind Rand. Marta finished in th place. Who finished in th place?

Solution

  
  2020-07-09 06:35:43

Question 6

Marley practices exactly one sport each day of the week. She runs three days a week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the day after running or swimming. Which day of the week does Marley swim?

Solution

  
  2020-07-09 06:35:43

Question 7

Consider the operation "minus the reciprocal of," defined by . What is ?

Solution

  
  2020-07-09 06:35:43

Question 8

The letter F shown below is rotated clockwise around the origin, then reflected in the -axis, and then rotated a half turn around the origin. What is the final image?

Solution

  
  2020-07-09 06:35:43

Question 9

The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius and center that lies in the first quadrant, the portion of the circle with radius and center that lies in the first quadrant, and the line segment from to . What is the area of the shark's fin falcata?

Solution

  
  2020-07-09 06:35:43

Question 10

What is the sign and units digit of the product of all the odd negative integers strictly greater than ?

Solution

  
  2020-07-09 06:35:43

Question 11

Among the positive integers less than , each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?

Solution

The one digit prime numbers are 2, 3, 5, and 7. So there are a total of 4*4=16 ways to choose a two digit number with both digits as primes and 4 ways to choose a one digit prime, for a total of 4+16=20 ways. Out of these 2, 3, 5, 7, 23, 37, 53, and 73 are prime. Thus the probability is 8/20.
  
  2017-02-08 04:51:40

Question 12

For how many integers is the point inside or on the circle of radius centered at ?

Solution

  
  2020-07-09 06:35:43

Question 13

The line forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?

Solution

If x=0, then y=12. If y is 0, then x=5. So the 3 vertices of the triangle are (0,0), (5,0), and (0,12).

Two of our altitudes are 5 and 12, and the hypotenuse can be calculated as 13. Since the area of the triangle is 30, so the final altitude is 30*2/13. The sum of our altitudes is 5+12+60/13 = 281/13
  
  2017-02-08 04:55:39

Question 14

Let , , and be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation ?

Solution

Expanding the equation and combining like terms results in

2x^2-(a+2b+c)x+(ab+bc)=0.

By Vieta's formula the sum of the roots is -[-(a+2b+c)] / 2=(a+2b+c)/2.

To maximize this expression we want b to be the largest, and from there we can assign the next highest values to a and c. So let b=9, a=8, and c=7. Then the answer is (8+18+7)/2=16.5.
  
  2017-02-08 04:58:25

Question 15

The town of Hamlet has people for each horse, sheep for each cow, and ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?

Solution

Let the number of people be p, horses be h, sheep be s, cows be c, and ducks be d.

We know
3h=p
4c=s
3p=d

Then the total number of people, horses, sheep, cows, and ducks may be written as 3h+h+4c+c+9h=13h+5c. Looking through the options, we see 47 is impossible to make for integer values of h and c. So the answer is 47.
  
  2017-02-08 05:00:09

Question 16

Al, Bill, and Cal will each randomly be assigned a whole number from to , inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?

Solution

  
  2020-07-09 06:35:43

Question 17

When the centers of the faces of the right rectangular prism shown below are joined to create an octahedron, what is the volume of the octahedron?

Solution

  
  2020-07-09 06:35:43

Question 18

Johann has fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?

Solution

  
  2020-07-09 06:35:43

Question 19

In , and . Squares and are constructed outside of the triangle. The points , and lie on a circle. What is the perimeter of the triangle?

Solution

  
  2020-07-09 06:35:43

Question 20

Erin the ant starts at a given corner of a cube and crawls along exactly edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?

Solution

  
  2020-07-09 06:35:43

Question 21

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of ?

Solution

  
  2020-07-09 06:35:43

Question 22

In the figure shown below, is a regular pentagon and . What is ?

Solution

  
  2020-07-09 06:35:43

Question 23

Let be a positive integer greater than 4 such that the decimal representation of ends in zeros and the decimal representation of ends in zeros. Let denote the sum of the four least possible values of . What is the sum of the digits of ?

Solution

  
  2020-07-09 06:35:43

Question 24

Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin facing to the east and walks one unit, arriving at . For , right after arriving at the point , if Aaron can turn left and walk one unit to an unvisited point , he does that. Otherwise, he walks one unit straight ahead to reach . Thus the sequence of points continues , and so on in a counterclockwise spiral pattern. What is ?

Solution

  
  2020-07-09 06:35:43

Question 25

A rectangular box measures , where and are integers and . The volume and surface area of the box are numerically equal. How many ordered triples are possible?

Solution

  
  2020-07-09 06:35:43

Answer Keys


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