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2007AMC12AProblemsandSolution

By Derrick Williams,2014-06-03 23:37
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2007AMC12AProblemsandSolution

2007 AMC 12A Problems and Solution

    Problem 1

    One ticket to a show costs at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?

Solution

    = the amount Pam spent = the amount Susan spent

    ;

    ;

    Pam pays 10 more dollars than Susan

    Problem 2

    An aquarium has a rectangular base that measures 100 cm by 40 cm and has a

    height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?

Solution

    The brick has volume . The base of the aquarium has area . For

    every inch the water rises, the volume increases by ; therefore, when the

    volume increases by , the water level rises

    Problem 3

    The larger of two consecutive odd integers is three times the smaller. What is their sum?

    Solution

    Solution 1 Let be the smaller term. Then

    ; Thus, the answer is

    Solution 2

    ; By trial and error, 1 and 3 work. 1+3=4.

    Problem 4

    Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90

    minutes at a speed of 4 mph. What was her overall average speed in miles per hour?

    Solution

    ;

    ;

    Problem 5

    Last year Mr. Jon Q. Public received an inheritance. He paid in federal taxes on

    the inheritance, and paid of what he had left in state taxes. He paid a total of

    $ for both taxes. How many dollars was his inheritance?

    Solution

    After paying his taxes, he has of the inheritance left. Since is

    of the inheritance, the whole inheritance is .

    Problem 6

    Triangles and are isosceles with and . Point

    is inside triangle , angle measures 40 degrees, and angle

    measures 140 degrees. What is the degree measure of angle ?

    Solution

    We angle chase, and find out that:

    ;

    ;

    ;

    Problem 7

    Let , and be five consecutive terms in an arithmetic sequence, and

    suppose that . Which of or can be found?

    Solution

    Let be the common difference between the terms. ;

    ;

    ;

    ;

    ;

    , so . But we can't find any more variables,

    because we don't know what is. So the answer is .

    Problem 8

    A star-polygon is drawn on a clock face by drawing a chord from each number to the

    fifth number counted clockwise from that number. That is, chords are drawn from

    12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree

    measure of the angle at each vertex in the star polygon?

Solution

We look at the angle between 12, 5, and 10. It subtends of the circle, or

    degrees (or you can see that the arc is of the right angle). Thus, the angle at each

    vertex is an inscribed angle subtending degrees, making the answer

Problem 9

    Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance

    from the stadium?

Solution

    Let the distance from Yan's initial position to the stadium be and the distance from Yan's initial position to home be . We are trying to find , and we have the following identity given by the problem:

Thus and the answer is

    Problem 10

    A triangle with side lengths in the ratio is inscribed in a circle with radius 3. What is the area of the triangle?

Solution

Since 3-4-5 is a Pythagorean triple, the triangle is a right triangle. Since the

    hypotenuse is a diameter of the circumcircle, the hypotenuse is . Then the

    other legs are and . The area is

    Problem 11

    A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let be the sum of all the

    terms in the sequence. What is the largest prime factor that always divides ?

Solution

    A given digit appears as the hundreds digit, the tens digit, and the units digit of a term the same number of times. Let be the sum of the units digits in all the terms. Then , so must be divisible by . To see that it need

    not be divisible by any larger prime, the sequence gives

    .

    Problem 12

    Integers and , not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that is even?

Solution

    The only times when is even is when and are of the same parity. The

    chance of being odd is , so it has a probability of being even.

    Therefore, the probability that will be even is .

Problem 13

    A piece of cheese is located at in a coordinate plane. A mouse is at and is running up the line . At the point the mouse starts getting farther from the cheese rather than closer to it. What is ?

Solution

    We are trying to find the foot of a perpendicular from to .

    Then the slope of the line that passes through the cheese and is the negative reciprocal of the slope of the line, or . Therefore, the line is . The

    point where and intersect is , and

    .

    Problem 14

    Let a, b, c, d, and e be distinct integers such that

What is ?

Solution

    If 45 is expressed as a product of five distinct integer factors, the absolute value of

    the product of any four it as least , so no factor can have an absolute value greater than 5. Thus the factors of the given expression are five of

    the integers . The product of all six of these is , so the factors are -3, -1, 1, 3, and 5. The corresponding values of a, b, c, d, and e are 9, 7,

    5, 3, and 1, and their sum is 25 (C).

Problem 15

    The set is augmented by a fifth element , not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of ?

Solution

    The median must either be or . Casework:

    ; Median is : Then and .

    ; Median is : Then and .

    ; Median is : Then and .

    All three cases are valid, so our solution is .

    Problem 16

    How many three-digit numbers are composed of three distinct digits such that one

    digit is the average of the other two?

Solution

    We can find the number of increasing arithmetic sequences of length 3 possible from 0 to 9, and then find all the possible permutations of these sequences.

    Common difference Sequences possible Number of sequences

    1 8

    2 6

    3 4

    4 2

    This gives us a total of sequences. There are to permute these, for a total of .

    However, we note that the conditions of the problem require three-digit numbers,

    and hence our numbers cannot start with zero. There are numbers which start with zero, so our answer is .

    Problem 17

    Suppose that and . What is ?

Solution

    We can make use the of the Pythagorean identities: square both equations and add them up:

    This is just the cosine difference identity, which simplifies to

Problem 18

    The polynomial has real coefficients, and

    What is

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