2018年美国数学竞赛 AMC 试题

2018 AIME I Problems
Problem 1
Let be the number of ordered pairs of
integers with and such that the
polynomial can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when is divided by .
Problem 2
The number can be written in base as , can be written in
base as , and can be written in base as , where . Find the base- representation of .
Problem 3
Kathy has red cards and green cards. She shuffles the cards and lays
out of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy,
but RRRGR will not. The probability that Kathy will be happy is ,
where and are relatively prime positive integers. Find . Problem 4
In and . Point lies strictly
between and on and point lies strictly between and on so
that . Then can be expressed in the form ,
where and are relatively prime positive integers. Find .
Problem 5
For each ordered pair of real numbers satisfying
there is a real number such that
Find the product of all possible values of .
Problem 6
Let be the number of complex numbers with the properties
that and is a real number. Find the remainder when is divided by .
Problem 7
A right hexagonal prism has height . The bases are regular hexagons with side length . Any of the vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).
Problem 8
Let be an equiangular hexagon such
that , and . Denote the diameter of the largest circle that fits inside the hexagon. Find .
Problem 9
Find the number of four-element subsets of with the property
that two distinct elements of a subset have a sum of , and two distinct elements of a subset have a sum of . For
example, and are two such subsets.
Problem 10
The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the
path , which has steps. Let be the number of paths with steps that begin and end at point . Find the remainder when is divided by .
Problem 11
Find the least positive integer such that when is written in base , its two right-most digits in base are .
Problem 12
For every subset of , let be the sum of the elements of , with defined to be . If is chosen at random among all
subsets of , the probability that is divisible by is , where and are relatively prime positive integers. Find .
Problem 13
Let have side lengths , , and .
Point lies in the interior of , and points and are the incenters
of and , respectively. Find the minimum possible area
of as varies along .
Problem 14
Let be a heptagon. A frog starts jumping at vertex . From any vertex of the heptagon except , the frog may jump to either of the two adjacent
vertices. When it reaches vertex , the frog stops and stays there. Find the number of distinct sequences of jumps of no more than jumps that end at .
Problem 15
David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , which can each be inscribed in a circle with radius . Let denote the measure of the acute angle made by the diagonals of quadrilateral , and define and similarly. Suppose
that , , and . All three quadrilaterals have the
same area , which can be written in the form , where and are relatively prime positive integers. Find .
2018 AMC 8 Problems
Problem 1
An amusement park has a collection of scale models, with ratio , of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number?
Problem 2
What is the value of the product
Problem 3
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
Problem 4
The twelve-sided figure shown has been drawn on graph paper. What is the area of the figure in ?
Problem 5
What is the value
of ?
Problem 6
On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?
Problem 7
The -digit number is divisible by . What is the remainder when this number is divided by ?
Problem 8
Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.
What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?
Problem 9
Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?
Problem 10
The of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?
Problem 11
Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.
If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?
Problem 12
The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?
Problem 13
Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?
Problem 14
Let be the greatest five-digit number whose digits have a product of . What is the sum of the digits of ?
Problem 15
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of square unit, then what is the area of the shaded region, in square units?
Problem 16
Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?
Problem 17
Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance
between their houses is miles, which is feet, and Bella covers feet with each step. How many steps will Bella take by the time she meets Ella?
Problem 18
How many positive factors does have?
Problem 19
In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?
Problem 20
In a point is on with and Point is
on so that and point is on so that What is the ratio of the area of to the area of
Problem 21
How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?
Problem 22
Point is the midpoint of side in square and meets diagonal at The area of quadrilateral is What is the area
of
Problem 23
From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?
Problem 24
In the cube with opposite vertices and and are the midpoints of edges and respectively. Let be the ratio of the area of the cross-section to the area of one of the faces of the cube. What is
Problem 25
How many perfect cubes lie between and , inclusive?
2018 AMC 10A Problems
Problem 1
What is the value of
Problem 2
Liliane has more soda than Jacqueline, and Alice has more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?
Liliane has more soda than Alice.
Liliane has more soda than Alice.
Liliane has more soda than Alice.
Liliane has more soda than Alice.
Liliane has more soda than Alice.
Problem 3
A unit of blood expires after seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?
Problem 4
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
Problem 5
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 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 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 6
Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
Problem 7
For how many (not necessarily positive) integer values of is the value of an integer?
Problem 8
Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?
Problem 9
All of the triangles in the diagram below are similar to iscoceles triangle , in
which . Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?
Problem 10
Suppose that real number satisfies. What is the value
of ?
Problem 11
When fair standard -sided die are thrown, the probability that the sum of the numbers on the top faces is can be written as, where is a positive integer. What is ?
Problem 12
How many ordered pairs of real numbers satisfy the following system of
equations?
Problem 13
A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?
Problem 14
What is the greatest integer less than or equal to
Problem 15
Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points and , as shown in the diagram. The distance can be written in the form , where and are relatively prime positive integers. What is ?
Problem 16
Right triangle has leg lengths and . Including and , how many line segments with integer length can be drawn from vertex to a point on hypotenuse ?
Problem 17
Let be a set of 6 integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible values of an element in
Problem 18
How many nonnegative integers can be written in the
form
where for ?
Problem 19
A number is randomly selected from the set , and a number is randomly selected from . What is the probability
that has a units digit of ?
Problem 20
A scanning code consists of a grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
Problem 21
Which of the following describes the set of values of for which the
curves and in the real -plane intersect at exactly points?
Problem 22
Let and be positive integers such
that , , ,
and . Which of the following must be a divisor of ?
Problem 23
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted
square so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?
Problem 24
Triangle with and has area . Let be the midpoint
of , and let be the midpoint of . The angle bisector
of intersects and at and , respectively. What is the area of quadrilateral ?
Problem 25
For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible
value of for which there are at least two values of such that ?
2018 AMC 10B Problems
Problem 1
Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?
Problem 2
Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?
Problem 3
In the expression each blank is to be filled in with one of the digits or with each digit being used once. How many different values can be obtained?
Problem 4
A three-dimensional rectangular box with dimensions , , and has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is ?
Problem 5
How many subsets of contain at least one prime number?
Problem 6
A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required?
Problem 7
In the figure below, congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let be the combined area of the small semicircles and be the area of the region inside the large semicircle but outside the small semicircles. The ratio is 1:18. What is ?
Problem 8
Sara makes a staircase out of toothpicks as shown:
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
Problem 9
The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What other sum occurs with the same probability ?
Problem 10
In the rectangular parallelepiped shown, , , and . Point is the midpoint of . What is the volume of the rectangular pyramid with base and apex ?
Problem 11
Which of the following expressions is never a prime number when is a prime number?
Problem 12
Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
Problem 13
How many of the first numbers in the sequence are divisible by ?
Problem 14
A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?
Problem 15
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point in the figure on the right. The box has base length and height . What is the area of the sheet of wrapping paper?
Problem 16
Let be a strictly increasing sequence of positive integers such
that What is the remainder
when is divided by ?
Problem 17
In rectangle , and . Points and lie on ,
points and lie on , points and lie on , and points and lie on so that and the convex octagon is equilateral. The length of a side of this octagon can be expressed in the form , where , , and are integers and is not divisible by the square of any prime. What is ?
Problem 18
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?
Problem 19
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
Problem 20
A function is defined recursively
by and for all
integers . What is ?
Problem 21
Mary chose an even -digit number . She wrote down all the divisors of in increasing order from
left to right: . At some moment Mary wrote as a divisor of . What is the smallest possible value of the next divisor written to the right of ?
Problem 22
Real numbers and are chosen independently and uniformly at random from the interval . Which of the following numbers is closest to the probability that and are the side lengths of an obtuse triangle?
Problem 23
How many ordered pairs of positive integers satisfy the
equation where denotes the greatest common divisor of and , and denotes their least common multiple?
Problem 24
Let be a regular hexagon with side length . Denote by , , and the midpoints of sides , , and , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of and ?
Problem 25
Let denote the greatest integer less than or equal to . How many real numbers satisfy the equation ?
2018 AMC 12A Problems
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 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 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 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 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 ?
Problem 7
For how many (not necessarily positive) integer values of is the value of an integer?
Problem 8
All of the triangles in the diagram below are similar to iscoceles triangle , in which
. Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?
Problem 9
Which of the following describes the largest subset of values of within the closed interval for
which for every between and , inclusive?
How many ordered pairs of real numbers satisfy the following system of equations?
Problem 11
A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?
Problem 12
Let be a set of 6 integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible value of an element in
Problem 13
How many nonnegative integers can be written in the form
where
for ?
Problem 14
The solutions to the equation , where is a positive real number other than
or , can be written as where and are relatively prime positive integers. What is ?
A scanning code consists of a grid of squares, with some of its squares colored black and the
rest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
Problem 16
Which of the following describes the set of values of for which the curves and
in the real -plane intersect at exactly points?
Problem 17
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square
so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?
Triangle with and has area . Let be the midpoint of
, and let be the midpoint of . The angle bisector of intersects and at
and , respectively. What is the area of quadrilateral ?
Problem 19
Let be the set of positive integers that have no prime factors other than , , or . The infinite sum
of the reciprocals of the elements of can be expressed as , where and are relatively prime
positive integers. What is ?
Problem 20
Triangle is an isosceles right triangle with . Let be the midpoint of
hypotenuse . Points and lie on sides and , respectively, so that
and is a cyclic quadrilateral. Given that triangle has area , the length can
be written as , where , , and are positive integers and is not divisible by the square of any prime. What is the value of ?
Problem 21
Which of the following polynomials has the greatest real root?
Problem 22
The solutions to the equations and where
form the vertices of a parallelogram in the complex plane. The area of this
parallelogram can be written in the form where and are positive integers
and neither nor is divisible by the square of any prime number. What is
Problem 23
In and Points and lie on sides
and respectively, so that Let and be the midpoints of segments
and respectively. What is the degree measure of the acute angle formed by lines
and
Problem 24
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the
numbers between and Armed with this information, what number should Carol choose to maximize her chance of winning?
Problem 25
For a positive integer and nonzero digits , , and , let be the -digit integer each of whose
digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be
the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?
2018 AMC 12B Problems
Problem 1
Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?
Problem 2
Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?
Problem 3
A line with slope 2 intersects a line with slope 6 at the point . What is the distance between the -intercepts of these two lines?
Problem 4
A circle has a chord of length , and the distance from the center of the circle to the chord is . What is the area of the circle?
Problem 5
How many subsets of contain at least one prime number?
Suppose cans of soda can be purchased from a vending machine for quarters. Which of the following expressions describes the number of cans of soda that can be purchased for dollars, where 1 dollar is worth 4 quarters?
Problem 7
What is the value of
Problem 8
Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
Problem 9
What is
Problem 10
A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?
Problem 11。