2007 AMC 12A Problems and Solution
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?
= the amount Pam spent = the amount Susan spent
Pam pays 10 more dollars than Susan
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?
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
The larger of two consecutive odd integers is three times the smaller. What is their sum?
Solution 1 Let be the smaller term. Then
; Thus, the answer is
; By trial and error, 1 and 3 work. 1+3=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?
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?
After paying his taxes, he has of the inheritance left. Since is
of the inheritance, the whole inheritance is .
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 ?
We angle chase, and find out that:
Let , and be five consecutive terms in an arithmetic sequence, and
suppose that . Which of or can be found?
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 .
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?
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
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?
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
A triangle with side lengths in the ratio is inscribed in a circle with radius 3. What is the area of the triangle?
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
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 ?
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
Integers and , not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that is even?
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 .
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 ?
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
Let a, b, c, d, and e be distinct integers such that
What is ?
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).
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 ?
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 .
How many three-digit numbers are composed of three distinct digits such that one
digit is the average of the other two?
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
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 .
Suppose that and . What is ?
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
The polynomial has real coefficients, and