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AMC 10 2002 B

Question 1

The ratio is:

Solution

  
  2020-07-09 06:36:03

Question 2

For the nonzero numbers and define Find .

Solution

  
  2020-07-09 06:36:03

Question 3

The arithmetic mean of the nine numbers in the set is a -digit number , all of whose digits are distinct. The number does not contain the digit

Solution

  
  2020-07-09 06:36:03

Question 4

What is the value of

when ?

Solution

  
  2020-07-09 06:36:03

Question 5

Circles of radius and are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.

Solution

  
  2020-07-09 06:36:03

Question 6

For how many positive integers is a prime number?

Solution

  
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Question 7

Let be a positive integer such that is an integer. Which of the following statements is not true?

Solution

  
  2020-07-09 06:36:03

Question 8

Suppose July of year has five Mondays. Which of the following must occur five times in the August of year ? (Note: Both months have days.)

Solution

  
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Question 9

Using the letters , , , , and , we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" occupies position

Solution

  
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Question 10

Suppose that and are nonzero real numbers, and that the equation has solutions and . what is the pair ?

Solution

  
  2020-07-09 06:36:03

Question 11

The product of three consecutive positive integers is times their sum. What is the sum of their squares?

Solution

Let the three consecutive positive integers be a-1, a, and a+1. So:
a(a-1)(a+1)=24a

(a-1)(a+1)=24

Since 24=4*6, we have a=5. The sum of the squares is 4^2+5^2+6^2=77
  
  2017-03-26 13:06:56

Question 12

For which of the following values of does the equation have no solution for ?

Solution

  
  2020-07-09 06:36:03

Question 13

Find the value(s) of such that is true for all values of .


Solution

  
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Question 14

The number is the square of a positive integer . In decimal representation, the sum of the digits of is

Solution

  
  2020-07-09 06:36:03

Question 15

The positive integers , , , and are all prime numbers. The sum of these four primes is

Solution 1

Review some basic even/odd rules here: https://www.homesweetlearning.com/resources/math/math910/numbers.html
  
  2017-01-05 18:30:28

Solution 2

Since A-B and A+B must have the same parity (both odd or both even), and since there is only one even prime number (number 2), it follows that A-B and A+B are both odd.

Since A+B is odd, one of A, B is odd and the other is even, ie prime even 2.

Since A+B > A > A-B, it follows that A is odd and B = 2.

So sum of 4 primes = 3 * A + 2

It cannot be even, cannot be divided by 3, cannot be divided by 5, cannot be divided by 7, so the only right choice is E.
  
  2017-01-05 18:26:34

Question 16

For how many integers is the square of an integer?


Solution

  
  2020-07-09 06:36:03

Question 17

A regular octagon has sides of length two. Find the area of .

Solution

  
  2020-07-09 06:36:03

Question 18

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

Solution

  
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Question 19

Suppose that is an arithmetic sequence with What is the value of

Solution

  
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Question 20

Let and be real numbers such that and Then is

Solution

  
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Question 21

Andy's lawn has twice as much area as Beth's lawn and three times as much as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first?

Solution

  
  2020-07-09 06:36:03

Question 22

Let be a right-angled triangle with . Let and be the midpoints of the legs and , respectively. Given and , find .

Solution

  
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Question 23

Let be a sequence of integers such that and for all positive integers and Then is

Solution

  
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Question 24

Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point vertical feet above the bottom?

Solution

  
  2020-07-09 06:36:03

Question 25

When is appended to a list of integers, the mean is increased by . When is appended to the enlarged list, the mean of the enlarged list is decreased by . How many integers were in the original list?

Solution

  
  2020-07-09 06:36:03

Answer Keys


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