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AMC 10 2005 B

Question 1

A scout troop buys candy bars at a price of five for $. They sell all the candy bars at a price of two for $. What was the profit, in dollars?

Solution

  
  2020-07-09 06:35:58

Question 2

A positive number has the property that of is . What is ?

Solution

  
  2020-07-09 06:35:58

Question 3

A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?

Solution

  
  2020-07-09 06:35:58

Question 4

For real numbers and , define . What is the value of

?

Solution

  
  2020-07-09 06:35:58

Question 5

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?

Solution

  
  2020-07-09 06:35:58

Question 6

At the beginning of the school year, Lisa's goal was to earn an A on at least of her quizzes for the year. She earned an A on of the first quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?

Solution

  
  2020-07-09 06:35:58

Question 7

A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?

Solution

  
  2020-07-09 06:35:58

Question 8

An -foot by -foot ???oor is tiled with square tiles of size foot by foot. Each tile has a pattern consisting of four white quarter circles of radius foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the ???oor are shaded?

Solution

  
  2020-07-09 06:35:58

Question 9

One fair die has faces , , , , , and another has faces , , , , , . The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?

Solution

  
  2020-07-09 06:35:58

Question 10

In , we have and . Suppose that is a point on line such that lies between and and . What is ?

Solution

  
  2020-07-09 06:35:58

Question 11

The first term of a sequence is . Each succeeding term is the sum of the cubes of the digits of the previous term. What is the term of the sequence?

Solution

  
  2020-07-09 06:35:58

Question 12

Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?

Solution

  
  2020-07-09 06:35:58

Question 13

How many numbers between and are integer multiples of or but not ?

Solution

  
  2020-07-09 06:35:58

Question 14

Equilateral has side length , is the midpoint of , and is the midpoint of . What is the area of ?

Solution

From description of the problem, we know MC=1 and CD=2; we also know angle MCD = 120.

Now, the area of the triangle will be given by the SAS SIN rule: https://www.homesweetlearning.com/resources/math/math910/geometry/areas.html
  
  2017-01-14 22:00:14

Question 15

An envelope contains eight bills: ones, fives, tens, and twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $ or more?

Solution

  
  2020-07-09 06:35:58

Question 16

The quadratic equation has roots that are twice those of , and none of , , and is zero. What is the value of ?

Solution

  
  2020-07-09 06:35:58

Question 17

Suppose that , , , and . What is ?

Solution

  
  2020-07-09 06:35:58

Question 18

All of David's telephone numbers have the form , where , , , , , , and are distinct digits and in increasing order, and none is either or . How many different telephone numbers can David have?

Solution

  
  2020-07-09 06:35:58

Question 19

On a certain math exam, of the students got points, got points, got points, got points, and the rest got points. What is the difference between the mean and the median score on this exam?

Solution

The scores are 70 (10%), 80 (25%), 85 (20%), 90 (15%), and 95 (30%). 

Suppose we have 20 students:

The scores are 70 (2), 80 (5), 85 (4), 90 (3), and 95 (6). 

The median is 85, and the mean is 86. So the diff is 1.
  
  2017-03-26 13:08:30

Question 20

What is the average (mean) of all -digit numbers that can be formed by using each of the digits , , , , and exactly once?

Solution

  
  2020-07-09 06:35:59

Question 21

Forty slips are placed into a hat, each bearing a number , , , , , , , , , or , with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let be the probability that all four slips bear the same number. Let be the probability that two of the slips bear a number and the other two bear a number . What is the value of ?

Solution

  
  2020-07-09 06:35:59

Question 22

For how many positive integers less than or equal to is evenly divisible by ?

Solution

  
  2020-07-09 06:35:59

Question 23

In trapezoid we have parallel to , as the midpoint of , and as the midpoint of . The area of is twice the area of . What is ?

Solution

  
  2020-07-09 06:35:59

Question 24

Let and be two-digit integers such that is obtained by reversing the digits of . The integers and satisfy for some positive integer . What is ?

Solution

  
  2020-07-09 06:35:59

Question 25

A subset of the set of integers from to , inclusive, has the property that no two elements of sum to . What is the maximum possible number of elements in ?

Solution

  
  2020-07-09 06:35:59

Answer Keys


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