2018年AMC 12A 真题：
Problem 1
A large urn contains 
balls, of which 
are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be 
? (No red balls are to be removed.)
Problem 2
While exploring a cave, Carl comes across a collection of 
-pound rocks worth 
each, 
-pound rocks worth 
each, and 
-pound rocks worth 
each. There are at least 
of each size. He can carry at most 
pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
Problem 3
How many ways can a student schedule 
mathematics courses -- algebra, geometry, and number theory -- in a 
-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other periods is of no concern here.)
Problem 4
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least miles away," Bob replied, "We are at most miles away." Charlie then remarked, "Actually the nearest town is at most miles away." It turned out that none of the three statements were true. Let 
be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?
Problem 5
What is the sum of all possible values of 
for which the polynomials 
and 
have a root in common?
Problem 6
For positive integers 
and 
such that 
, both the mean and the median of the set 
are equal to . What is 
?
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degrees warmer than St. Louis. At 
the temperature in Minneapolis has fallen by 
degrees. What is the product of all possible values of 
. Which of the following is equal to 
, not necessarily in the simplest form, special if 
and 
are positive integers whose sum is 
. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
is 
. What is the sum of the digits of the greatest prime number that is a divisor of 
?
?
index card. If she shortens the length of one side of this card by 
miles per hour. Route B is 
miles per hour, except for a 
-mile stretch in a school zone where his average speed is 
is prime for only one digit 
What is 
equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in 
equal leaps. The telephone poles are evenly spaced, and the 
st pole along this road is exactly one mile (
feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
lies on the opposite half-plane determined by line 
from point 
so that 
. Point 
lies on 
so that 
, and 
is a square. What is the degree measure of 
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true, where 
It is never true.
It is true if and only if 
.
It is true if and only if 
.
It is true if and only if 
It is always true.
. One of the two numbers is divisible by 
. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
snakes, 
by the repeating decimal,![\[\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},\]](https://latex.artofproblemsolving.com/d/8/a/d8a2b2bed7d27aa171aee4df068bd644bdcb9bb7.png)
where 
Later he found that his answer is 
less than the correct answer. What is the 
. When 
. How many cards were in the deck originally?
satisfy 
students are wearing blue shirts, and another 
students are wearing yellow shirts. The 
students are assigned into 
of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?
![\[2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.\]](https://latex.artofproblemsolving.com/9/f/2/9f2bfdf62a315bd20a6a692df961e22f22f7611d.png)
What is the value of 
, and the afternoon class's mean score is 
. The ratio of the number of students in the morning class to the number of students in the afternoon class is 
. What is the mean of the score of all the students?
in the 
-plane is first rotated counterclockwise by 
around the point 
and then reflected about the line 
. The image of 
after these two transformations is at 
. What is 
and height 
is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of 
. What is the height in centimeters of the water in the cylinder?
?
?
cents less. Ben paid 
more than David. How much did they spend in the bagel store together?
with 
by![\[\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor\]](https://latex.artofproblemsolving.com/9/5/c/95c21b1ce04c5f3c86f558991d39aa35a13bca21.png)
where 
denotes the greatest integer less than or equal to 
. What is the value of 
?
. Which of the following could be the volume of the box?
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. After he graded Payton's test, the test average became 
. What was Payton's score on the test?
of a can of cat food every morning and 
of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing 
each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs 
for her to make. In dollars, what is her profit for the day?
of the students scored 
scored 
scored 
points, and the rest scored 
days. At this rate, how many gallons of milk will 
days?
cups of sugar. Unfortunately, her measuring cup holds only 
, and the area of 
is 
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, Dorothy paid 
, and Sammy paid 
. In order to share costs equally, Tom gave Sammy 
dollars, and Dorothy gave Sammy 
?
?
degrees higher than the low temperature, and the average of the high and the low temperatures was 
. In degrees, what was the low temperature in Lincoln that day?
, 
is the 
number counted. When counting backwards from 
number counted. What is 
?
