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AMC 10 2005 A

Question 1

While eating out, Mike and Joe each tipped their server dollars. Mike tipped of his bill and Joe tipped of his bill. What was the difference, in dollars between their bills?

Solution

  
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Question 2

For each pair of real numbers , define the operation as

.

What is the value of ?

Solution

  
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Question 3

The equations and have the same solution . What is the value of ?

Solution

  
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Question 4

A rectangle with a diagonal of length is twice as long as it is wide. What is the area of the rectangle?

Solution

  
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Question 5

A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?

Solution

  
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Question 6

The average (mean) of numbers is , and the average of other numbers is . What is the average of all numbers?

Solution

  
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Question 7

Josh and Mike live miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?

Solution

  
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Question 8

In the figure, the length of side of square is and . What is the area of the inner square ?

Solution

  
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Question 9

Three tiles are marked and two other tiles are marked . The five tiles are randomly arranged in a row. What is the probability that the arrangement reads ?

Solution

  
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Question 10

There are two values of for which the equation has only one solution for . What is the sum of those values of ?

Solution

  
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Question 11

A wooden cube units on a side is painted red on all six faces and then cut into unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is ?

Solution

Since there are n^2 little faces on each face of the big wooden cube, there are 6*n^2 little faces painted red.

Since each unit cube has 6 faces, and there are n^3 unit cubes, there are 6*n^3 little faces total.

Since one-fourth of the little faces are painted red, we have:

6n^2 / 6n^3 = 1/4

Solving above, we have n=4
  
  2017-02-08 03:57:33

Question 12

The figure shown is called a trefoil and is constructed by drawing circular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length ?

Solution

The area of the trefoil is equal to the area of a small equilateral triangle plus the area of four 60 degree sectors with a radius of 2/2=1 minus the area of a small equilateral triangle.
  
  2017-01-21 22:11:58

Question 13

How many positive integers satisfy the following condition:

?

Solution

From what is given, we have:

130n > n^2 > 16

Solving each part separately:

n^2 > 16 -> n > 4

130n > n^2 -> 130 > n

Therefore the answer is the number of positive integers over the interval (4,130) which is 125.
  
  2017-01-21 22:16:01

Question 14

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

Solution

We could note that the middle digit is uniquely defined by the first and third digits, since it is half of their sum. This also means that the sum of the first and third digits must be even. Since even numbers are formed either by adding two odd numbers or two even numbers, we can split our problem into 2 cases:

If both the first digit and the last digit are odd, then we have 1, 3, 5, 7, or 9 as choices for each of these digits, and there are 5*5=25 numbers in this case.

If both the first and last digits are even, then we have 2, 4, 6, 8 as our choices for the first digit and 0, 2, 4, 6, 8 for the third digit. There are 4*5=20 numbers here.

The total number, then, is 20+25=45
  
  2017-01-21 22:19:41

Question 15

How many positive cubes divide  ?

Solution

3! * 5! * 7! = (3 * 2 * 1) * (5 * 4 * 3 * 2 * 1) * (7 * 6 * 5 * 4 * 3 * 2 * 1) = 2^8 * 3^4 * 5^2 * 7^1

Therefore, a perfect cube that divides 3!*5!*7! must be in the form 2^a*3^b*5^c*7^d where a, b, c, and d are nonnegative multiples of 3 that are less than or equal to 8, 4, 2 and 1, respectively.

So:

a in {0,3,6} (3 possibilities)

b in {0,3} (2 possibilities)

c in {0} (1 possibility)

d in {0} (1 possibility)


So the number of perfect cubes that divide 3!*5!*7! is 3 * 2 * 1 * 1 = 6
  
  2017-01-21 22:26:37

Question 16

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is . How many two-digit numbers have this property?

Solution

Let the number be 10a+b where a and b are the tens and units digits of the number.

So (10a+b)-(a+b)=9a must have a units digit of 6

This is only possible if 9a=36, so a=4 is the only way this can be true.

So the numbers that have this property are 40, 41, 42, 43, 44, 45, 46, 47, 48, 49.

Therefore the answer is 10
  
  2017-01-21 22:28:49

Question 17

In the five-sided star shown, the letters , , , , and are replaced by the numbers , , , , and , although not necessarily in this order. The sums of the numbers at the ends of the line segments , , , , and form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence?

Solution

  
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Question 18

Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?

Solution

  
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Question 19

Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point from the line on which the bases of the original squares were placed?

Solution

  
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Question 20

An equiangular octagon has four sides of length 1 and four sides of length , arranged so that no two consecutive sides have the same length. What is the area of the octagon?

Solution

  
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Question 21

For how many positive integers does evenly divide ?

Solution

Arithmetic sequence: https://www.homesweetlearning.com/resources/math/math910/numbers/arithmetic_sequence.html

The sum of x1 + x2 + x3 + ... + xn is: ( x1 + xn ) * n / 2

6n/(( x1 + xn ) * n / 2) = 12/(n+1). When n=1,2,3,5,11 will it be an integer.
  
  2017-01-07 17:04:30

Question 22

Let be the set of the smallest positive multiples of , and let be the set of the smallest positive multiples of . How many elements are common to and ?

Solution

S = {4, 8, 12, ..., 4*2005} = {4, 8, 12, ..., 8020}
T = {6, 12, 18, 24, ..., 6*2005} = {6, 12, 18, 24, ..., 12030}

Least common multiples of 4 and 6 is 12.
All multiples of 12 in S are also in T.

In multiples of 4, every 3rd one is a multiple of 12. So in S, we have 2005/3 = 668 multiples of 12.
  
  2017-01-07 18:25:42

Question 23

Let be a diameter of a circle and let be a point on with . Let and be points on the circle such that and is a second diameter. What is the ratio of the area of to the area of ?

Solution

  
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Question 24

For each positive integer , let denote the greatest prime factor of . For how many positive integers is it true that both and ?

Solution

  
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Question 25

In we have , , and . Points and are on and respectively, with and . What is the ratio of the area of triangle to the area of the quadrilateral ?

Solution

  
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Answer Keys


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