2018 AMC 10A试题及答案
2018年美国数学竞赛 AMC 试题
2018 AIME I ProblemsProblem 1Let be the number of ordered pairs ofintegers with and such that thepolynomial can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when is divided by .Problem 2The number can be written in base as , can be written inbase as , and can be written in base as , where . Find the base- representation of .Problem 3Kathy has red cards and green cards. She shuffles the cards and laysout 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 4In and . Point lies strictlybetween and on and point lies strictly between and on sothat . Then can be expressed in the form ,where and are relatively prime positive integers. Find .Problem 5For each ordered pair of real numbers satisfyingthere is a real number such thatFind the product of all possible values of .Problem 6Let be the number of complex numbers with the propertiesthat and is a real number. Find the remainder when is divided by .Problem 7A 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 8Let be an equiangular hexagon suchthat , and . Denote the diameter of the largest circle that fits inside the hexagon. Find .Problem 9Find the number of four-element subsets of with the propertythat two distinct elements of a subset have a sum of , and two distinct elements of a subset have a sum of . Forexample, and are two such subsets.Problem 10The 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 thepath , 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 11Find the least positive integer such that when is written in base , its two right-most digits in base are .Problem 12For every subset of , let be the sum of the elements of , with defined to be . If is chosen at random among allsubsets of , the probability that is divisible by is , where and are relatively prime positive integers. Find .Problem 13Let have side lengths , , and .Point lies in the interior of , and points and are the incentersof and , respectively. Find the minimum possible areaof as varies along .Problem 14Let 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 adjacentvertices. 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 15David 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. Supposethat , , and . All three quadrilaterals have thesame area , which can be written in the form , where and are relatively prime positive integers. Find .2018 AMC 8 ProblemsProblem 1An 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 2What is the value of the productProblem 3Students 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 4The twelve-sided figure shown has been drawn on graph paper. What is the area of the figure in ?Problem 5What is the valueof ?Problem 6On 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 7The -digit number is divisible by . What is the remainder when this number is divided by ?Problem 8Mr. 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 9Tyler 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 10The 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 11Abby, 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 12The 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 13Laila 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 14Let be the greatest five-digit number whose digits have a product of . What is the sum of the digits of ?Problem 15In 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 16Professor 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 17Bella 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 distancebetween 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 18How many positive factors does have?Problem 19In 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 20In a point is on with and Point ison so that and point is on so that What is the ratio of the area of to the area ofProblem 21How 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 22Point is the midpoint of side in square and meets diagonal at The area of quadrilateral is What is the areaofProblem 23From 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 24In 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 isProblem 25How many perfect cubes lie between and , inclusive?2018 AMC 10A ProblemsProblem 1What is the value ofProblem 2Liliane 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 3A 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 4How 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 5Alice, 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 6Sangho 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 7For how many (not necessarily positive) integer values of is the value of an integer?Problem 8Joe 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 9All of the triangles in the diagram below are similar to iscoceles triangle , inwhich . Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?Problem 10Suppose that real number satisfies. What is the valueof ?Problem 11When 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 12How many ordered pairs of real numbers satisfy the following system ofequations?Problem 13A 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 14What is the greatest integer less than or equal toProblem 15Two 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 16Right 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 17Let 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 inProblem 18How many nonnegative integers can be written in theformwhere for ?Problem 19A number is randomly selected from the set , and a number is randomly selected from . What is the probabilitythat has a units digit of ?Problem 20A 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 21Which of the following describes the set of values of for which thecurves and in the real -plane intersect at exactly points?Problem 22Let and be positive integers suchthat , , ,and . Which of the following must be a divisor of ?Problem 23Farmer 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 unplantedsquare 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 24Triangle with and has area . Let be the midpointof , and let be the midpoint of . The angle bisectorof intersects and at and , respectively. What is the area of quadrilateral ?Problem 25For 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 possiblevalue of for which there are at least two values of such that ?2018 AMC 10B ProblemsProblem 1Kate 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 2Sam 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 3In 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 4A 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 5How many subsets of contain at least one prime number?Problem 6A 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 7In 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 8Sara 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 9The 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 10In the rectangular parallelepiped shown, , , and . Point is the midpoint of . What is the volume of the rectangular pyramid with base and apex ?Problem 11Which of the following expressions is never a prime number when is a prime number?Problem 12Line 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 13How many of the first numbers in the sequence are divisible by ?Problem 14A 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 15A 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 16Let be a strictly increasing sequence of positive integers suchthat What is the remainderwhen is divided by ?Problem 17In 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 18Three 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 19Joey 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 20A function is defined recursivelyby and for allintegers . What is ?Problem 21Mary chose an even -digit number . She wrote down all the divisors of in increasing order fromleft 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 22Real 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 23How many ordered pairs of positive integers satisfy theequation where denotes the greatest common divisor of and , and denotes their least common multiple?Problem 24Let 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 25Let denote the greatest integer less than or equal to . How many real numbers satisfy the equation ?2018 AMC 12A ProblemsProblem 1A 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 2While exploring a cave, Carl comes across a collection of -pound rocks worth each, -poundrocks 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 3How 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 4Alice, 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 5What is the sum of all possible values of for which the polynomials andhave a root in common?Problem 6For positive integers and such that , both the mean and the median ofthe set are equal to . What is ?Problem 7For how many (not necessarily positive) integer values of is the value of an integer?Problem 8All 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 9Which of the following describes the largest subset of values of within the closed interval forwhich for every between and , inclusive?How many ordered pairs of real numbers satisfy the following system of equations?Problem 11A 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 12Let 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 inProblem 13How many nonnegative integers can be written in the formwherefor ?Problem 14The solutions to the equation , where is a positive real number other thanor , 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 therest 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 16Which of the following describes the set of values of for which the curves andin the real -plane intersect at exactly points?Problem 17Farmer 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 squareso 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 atand , respectively. What is the area of quadrilateral ?Problem 19Let be the set of positive integers that have no prime factors other than , , or . The infinite sumof the reciprocals of the elements of can be expressed as , where and are relatively primepositive integers. What is ?Problem 20Triangle is an isosceles right triangle with . Let be the midpoint ofhypotenuse . Points and lie on sides and , respectively, so thatand is a cyclic quadrilateral. Given that triangle has area , the length canbe written as , where , , and are positive integers and is not divisible by the square of any prime. What is the value of ?Problem 21Which of the following polynomials has the greatest real root?Problem 22The solutions to the equations and whereform the vertices of a parallelogram in the complex plane. The area of thisparallelogram can be written in the form where and are positive integersand neither nor is divisible by the square of any prime number. What isProblem 23In and Points and lie on sidesand respectively, so that Let and be the midpoints of segmentsand respectively. What is the degree measure of the acute angle formed by linesandProblem 24Alice, 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 thenumbers between and Armed with this information, what number should Carol choose to maximize her chance of winning?Problem 25For a positive integer and nonzero digits , , and , let be the -digit integer each of whosedigits is equal to ; let be the -digit integer each of whose digits is equal to , and let bethe -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 ProblemsProblem 1Kate 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 2Sam 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 3A 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 4A 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 5How 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 7What is the value ofProblem 8Line 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 9What isProblem 10A 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。
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2010-2015年AMC 10A和B竞赛真题及答案（英文版）
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2010年AMC 10Aห้องสมุดไป่ตู้赛真题及答案（英文版）
2010年AMC 10B竞赛真题及答案（英文版）
2011年AMC 10A竞赛真题及答案（英文版）
2011年AMC 10B竞赛真题及答案（英文版）
Problem 4
2012年AMC 10A竞赛真题及答案（英文版）
go去wentgone
2012年AMC 10B竞赛真题及答案（英文版）
get得到gotgot
sink下沉sank / sunksunk / sunken
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2013年AMC 10A竞赛真题及答案（英文版）
mistake误认mistookmistaken
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AMC10A 2018真题和答案
What is the value of Liliane has more soda than Jacqueline, and Alice has more soda than Jacqueline. What is the relationship between theamounts of soda that Liliane and Alice have? Liliane hasmore soda than Alice. Liliane hasmore soda than Alice. Liliane hasmore soda than Alice. Liliane hasmore soda than Alice. Liliane has more soda than Alice. A unit of blood expires after seconds. Yasin donates a unit of blood at noon of January 1. On what daydoes his unit of blood expire?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 noconcern here.)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 6miles 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 followingintervals is the set of all possible values of ?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 thatof the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?2018 AMC 10A ProblemsProblem 1Problem 2Problem 3Problem 4Problem 5Problem 6For how many (not necessarily positive) integer values of is the value ofan integer?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?All of the triangles in the diagram below are similar to isosceles triangle , in which . Each of the 7 smallesttriangles has area 1, andhas area 40. What is the area of trapezoid?Suppose that real numbersatisfies . What is the value of ?Whenfair 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 7Problem 8Problem 9Problem 10Problem 11How many ordered pairs of real numberssatisfy the following system of equations? 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 ininches of the crease?What is the greatest integer less than or equal to 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 12Problem 13Problem 14Problem 15Right triangle has leg lengths and . Includingand , how many line segments with integer length can be drawn from vertex to a point on hypotenuse ?Let be a set of 6 integers taken from with the property that ifandare elements of with , thenis not a multiple of . What is the least possible value of an element in How many nonnegative integers can be written in the formwhere for ?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 ?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 Problem 16Problem 17Problem 18Problem 19Problem 20line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanningcodes?Which of the following describes the set of values offor which the curves and in the real -plane intersect at exactly points?Let andbe positive integers such that , , , and. Which of the following must be a divisor of?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?Trianglewith and has area . Let be the midpoint of, and letbe the midpoint of . The angle bisector of intersects and at and , respectively. What is the area of quadrilateral ?For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let bethe -digit integer each of whose digits is equal to , and letbe 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?Problem 21Problem 22Problem 23Problem 24Problem 252018 AMC 10A Answer Key1. B2. A3. E4. E5. D6. B7. E8. C9. E10. A11. E12. C13. D14. A15. D16. D17. C18. D19. E20. B21. E22. D23. D24. D25. D。
2018年全国高中数学联合竞赛一试参考答案(A卷)
2018年全国高中数学联合竞赛一试（A 卷）参考答案及评分标准说明：1. 评阅试卷时，请依据本评分标准. 填空题只设8分和0分两档；其他各题的评阅，请严格按照本评分标准的评分档次给分，不得增加其他中间档次.2. 如果考生的解答方法和本解答不同，只要思路合理、步骤正确，在评卷时可参考本评分标准适当划分档次评分，解答题中第9小题4分为一个档次，第10、11小题5分为一个档次，不得增加其他中间档次．一、填空题：本大题共8小题，每小题8分，满分64分．1. 设集合 1,2,3,,99,2,2A B x x A C x x A ，则B C 的元素个数为 ．答案：24．解：由条件知， 13992,4,6,,198,1,,2,,2,4,6,,48222B C，故B C 的元素个数为24．2. 设点P 到平面的距离为，点Q 在平面 上，使得直线PQ 与 所成角不小于30 且不大于60 ，则这样的点Q 所构成的区域的面积为 ．答案：8 ．解：设点P 在平面 上的射影为O ．由条件知，tan OP OQP OQ ，即[1,3]OQ ，故所求的区域面积为22318 ．3. 将1,2,3,4,5,6随机排成一行，记为,,,,,a b c d e f ，则abc def +是偶数的概率为 ．答案：910．解：先考虑abc def +为奇数的情况，此时,abc def 一奇一偶，若abc 为奇数，则,,a b c 为1,3,5的排列，进而,,d e f 为2,4,6的排列，这样有3!3!36×=种情况，由对称性可知，使abc def +为奇数的情况数为36272×=种．从而abc def +为偶数的概率为72729116!72010−=−=．4. 在平面直角坐标系xOy 中，椭圆2222:1(0)x y C a b a b的左、右焦点分别是1F 、2F ，椭圆C 的弦ST 与UV 分别平行于x 轴与y 轴，且相交于点P ．已知线段,,,PU PS PV PT 的长分别为1,2,3,6，则12PF F 的面积为 ．答案解：由对称性，不妨设(,)P P P x y 在第一象限，则由条件知112,122P P x PT PS y PV PU ，即(2,1)P ．进而由1,2P x PU PS 得(2,2),(4,1)U S ，代入椭圆C 的方程知2222111144161a b a b，解得2220,5a b ．从而121212PF F P P S F F y y ．5. 设()f x 是定义在R 上的以2为周期的偶函数，在区间[0,1]上严格递减，且满足()1,(2)2f f ，则不等式组12,1()2x f x的解集为 ． 答案：[2,82] ．解：由()f x 为偶函数及在[0,1]上严格递减知，()f x 在[1,0] 上严格递增，再结合()f x 以2为周期可知，[1,2]是()f x 的严格递增区间．注意到(2)()1,(82)(2)(2)2f f f f f ，所以1()2(2)()(82)f x f f x f ，而12822 ，故原不等式组成立当且仅当[2,82]x ．6. 设复数z 满足1z ，使得关于x 的方程2220zx zx 有实根，则这样的复数z 的和为 ．答案：32．解：设22i (,,1)R z a b a b a b ．将原方程改为2(i)2(i)20a b x a b x ，分离实部与虚部后等价于2220ax ax ，① 220bx bx ．②若0b ，则21a ，但当1a 时，①无实数解，从而1a ，此时存在实数1x 1z 满足条件．若0b ，则由②知{0,2}x，但显然0x 不满足①，故只能是2x ，代入①解得14a ，进而b，相应有z ．综上，满足条件的所有复数z 之和为312．7. 设O 为ABC 的外心，若2AO AB AC，则sin BAC 的值为 ．答案 解：不失一般性，设ABC 的外接圆半径2R ．由条件知，2AC AO AB BO，①故112AC BO ．取AC 的中点M ，则OM AC ，结合①知OM BO ，且B 与A 位于直线OM 的同侧．于是1cos cos(90)sin 4MC BOC MOC MOC OC ． 在BOC 中，由余弦定理得BC ，进而在ABC中，由正弦定理得sin 2BC BAC R． 8. 设整数数列1210,,,a a a 满足1012853,2a a a a a ，且1{1,2},1,2,,9i i i a a a i ，则这样的数列的个数为 ．答案：80．解：设1{1,2}(1,2,,9)i i i b a a i ，则有11011292a a a b b b ，① 2345285567b b b a a a a b b b ．②用t 表示234,,b b b 中值为2的项数．由②知，t 也是567,,b b b 中值为2的项数，其中{0,1,2,3}t ．因此237,,,b b b 的取法数为021222323333(C )(C )(C )(C )20 ．取定237,,,b b b 后，任意指定89,b b 的值，有224 种方式．最后由①知，应取1{1,2}b 使得129b b b 为偶数，这样的1b 的取法是唯一的，并且确定了整数1a 的值，进而数列129,,,b b b 唯一对应一个满足条件的数列1210,,,a a a ．综上可知，满足条件的数列的个数为20480 ．二、解答题：本大题共3小题，满分56分．解答应写出文字说明、证明过程或演算步骤．9.（本题满分16分）已知定义在R 上的函数()f x 为3log 1,09,()49.x x f x x设,,a b c 是三个互不相同的实数，满足()()()f a f b f c ，求abc 的取值范围．解：不妨假设a b c ．由于()f x 在(0,3]上严格递减，在[3,9]上严格递增，在[9,) 上严格递减，且(3)0,(9)1f f ，故结合图像可知(0,3)a ，(3,9)b ，(9,)c ，并且()()()(0,1)f a f b f c ． …………………4分由()()f a f b 得331log log 1a b ，即33log log 2a b ，因此239ab ．于是9abc c ． …………………8分又0()41f c ， …………………12分故(9,16)c ．进而9(81,144)abc c ．所以，abc 的取值范围是(81,144)． …………………16分注：对任意的(81,144)r ，取09rc =，则0(9,16)c ∈，从而0()(0,1)f c ∈．过点00(,())c f c 作平行于x 轴的直线l ，则l 与()f x 的图像另有两个交点(,())a f a ，(,())b f b （其中(0,3),(3,9)a b ），满足()()()f a f b f c ，并且9ab ，从而abc r =．10.（本题满分20分）已知实数列123,,,a a a 满足：对任意正整数n ，有(2)1n n n a S a ，其中n S 表示数列的前n 项和．证明：(1) 对任意正整数n ，有n a(2) 对任意正整数n ，有11n n a a ．证明：(1) 约定00S ．由条件知，对任意正整数n ，有221111(2)()()n n n n n n n n n a S a S S S S S S ，从而220n S n S n ，即n S （当0n 时亦成立）． …………………5分显然，1n n n a S S ． …………………10分 (2) 仅需考虑1,n n a a 同号的情况．不失一般性，可设1,n n a a 均为正（否则将数列各项同时变为相反数，仍满足条件），则11n n n S S S ，故必有1n n S S ，此时1n n a a从而11n n a a ．…………………20分11.（本题满分20分）在平面直角坐标系xOy 中，设AB 是抛物线24y x 的过点(1,0)F 的弦，AOB 的外接圆交抛物线于点P （不同于点,,O A B ）．若PF 平分APB ，求PF 的所有可能值．解：设222123123,,,,,444y y y A y B y P y，由条件知123,,y y y 两两不等且非零． 设直线AB 的方程为1x ty ，与抛物线方程联立可得2440y ty ，故124y y ． ① 注意到AOB 的外接圆过点O ，可设该圆的方程为220x y dx ey ，与24y x 联立得，4210164y d y ey ．该四次方程有123,,,0y y y y 这四个不同的实根，故由韦达定理得12300y y y ，从而312()y y y ．②…………………5分因PF 平分APB ，由角平分线定理知，12PA FA yPB FB y ，结合①、②，有2222312222231212112122222222222321222132()()16(2)44()16(2)()44y y y y y y y y y PA yy PB y y y y y y y y y2222422122122224212112(8)16(416)64192(8)16(416)64192y y y y y y y y y y ， ………………10分 即62226222112122126419264192y y y y y y y y ，故 224224121122()(192)0y y y y y y ．当2212y y 时，21y y ，故30y ，此时P 与O 重合，与条件不符． 当422411221920y y y y 时，注意到①，有22221212()192()208y y y y ． …………………15分因22121282y y y y ，故满足①以及2212y y 的实数12,y y 存在，对应可得满足条件的点,A B ．此时，结合①、②知222231212()4411444y y y y y PF ．…………………20分。
2018年全国高中数学联赛A试题+答案
2018年全国高中数学联合竞赛一试（A 卷）参考答案及评分标准说明：1. 评阅试卷时，请依据本评分标准. 填空题只设8分和0分两档；其他各题的评阅，请严格按照本评分标准的评分档次给分，不得增加其他中间档次.2. 如果考生的解答方法和本解答不同，只要思路合理、步骤正确，在评卷时可参考本评分标准适当划分档次评分，解答题中第9小题4分为一个档次，第10、11小题5分为一个档次，不得增加其他中间档次．一、填空题：本大题共8小题，每小题8分，满分64分．1. 设集合 1,2,3,,99,2,2A B x x A C x x A ，则B C 的元素个数为 ．答案：24．解：由条件知， 13992,4,6,,198,1,,2,,2,4,6,,48222B C，故B C 的元素个数为24．2. 设点P 到平面的距离为，点Q 在平面 上，使得直线PQ 与 所成角不小于30 且不大于60 ，则这样的点Q 所构成的区域的面积为 ．答案：8 ．解：设点P 在平面 上的射影为O ．由条件知，tan OP OQP OQ ，即[1,3]OQ ，故所求的区域面积为22318 ．3. 将1,2,3,4,5,6随机排成一行，记为,,,,,a b c d e f ，则abc def +是偶数的概率为 ．答案：910．解：先考虑abc def +为奇数的情况，此时,abc def 一奇一偶，若abc 为奇数，则,,a b c 为1,3,5的排列，进而,,d e f 为2,4,6的排列，这样有3!3!36×=种情况，由对称性可知，使abc def +为奇数的情况数为36272×=种．从而abc def +为偶数的概率为72729116!72010−=−=．4. 在平面直角坐标系xOy 中，椭圆2222:1(0)x y C a b a b的左、右焦点分别是1F 、2F ，椭圆C 的弦ST 与UV 分别平行于x 轴与y 轴，且相交于点P ．已知线段,,,PU PS PV PT 的长分别为1,2,3,6，则12PF F 的面积为 ．答案解：由对称性，不妨设(,)P P P x y 在第一象限，则由条件知112,122P P x PT PS y PV PU ，即(2,1)P ．进而由1,2P x PU PS 得(2,2),(4,1)U S ，代入椭圆C 的方程知2222111144161a b a b，解得2220,5a b ．从而121212PF F P P S F F y y ．5. 设()f x 是定义在R 上的以2为周期的偶函数，在区间[0,1]上严格递减，且满足()1,(2)2f f ，则不等式组12,1()2x f x的解集为 ． 答案：[2,82] ．解：由()f x 为偶函数及在[0,1]上严格递减知，()f x 在[1,0] 上严格递增，再结合()f x 以2为周期可知，[1,2]是()f x 的严格递增区间．注意到(2)()1,(82)(2)(2)2f f f f f ，所以1()2(2)()(82)f x f f x f ，而12822 ，故原不等式组成立当且仅当[2,82]x ．6. 设复数z 满足1z ，使得关于x 的方程2220zx zx 有实根，则这样的复数z 的和为 ．答案：32．解：设22i (,,1)R z a b a b a b ．将原方程改为2(i)2(i)20a b x a b x ，分离实部与虚部后等价于2220ax ax ，① 220bx bx ．②若0b ，则21a ，但当1a 时，①无实数解，从而1a ，此时存在实数1x 1z 满足条件．若0b ，则由②知{0,2}x，但显然0x 不满足①，故只能是2x ，代入①解得14a ，进而bz ．综上，满足条件的所有复数z 之和为312．7. 设O 为ABC 的外心，若2AO AB AC，则sin BAC 的值为 ．答案 解：不失一般性，设ABC 的外接圆半径2R ．由条件知，2AC AO AB BO，①故112AC BO ．取AC 的中点M ，则OM AC ，结合①知OM BO ，且B 与A 位于直线OM 的同侧．于是1cos cos(90)sin 4MCBOC MOC MOC OC． 在BOC 中，由余弦定理得BC ，进而在ABC中，由正弦定理得sin 2BC BAC R． 8. 设整数数列1210,,,a a a 满足1012853,2a a a a a ，且1{1,2},1,2,,9i i i a a a i ，则这样的数列的个数为 ．答案：80．解：设1{1,2}(1,2,,9)i i i b a a i ，则有11011292a a a b b b ，① 2345285567b b b a a a a b b b ．②用t 表示234,,b b b 中值为2的项数．由②知，t 也是567,,b b b 中值为2的项数，其中{0,1,2,3}t ．因此237,,,b b b 的取法数为021222323333(C )(C )(C )(C )20 ．取定237,,,b b b 后，任意指定89,b b 的值，有224 种方式．最后由①知，应取1{1,2}b 使得129b b b 为偶数，这样的1b 的取法是唯一的，并且确定了整数1a 的值，进而数列129,,,b b b 唯一对应一个满足条件的数列1210,,,a a a ．综上可知，满足条件的数列的个数为20480 ．二、解答题：本大题共3小题，满分56分．解答应写出文字说明、证明过程或演算步骤．9.（本题满分16分）已知定义在R 上的函数()f x 为3log 1,09,()49.x x f x x设,,a b c 是三个互不相同的实数，满足()()()f a f b f c ，求abc 的取值范围．解：不妨假设a b c ．由于()f x 在(0,3]上严格递减，在[3,9]上严格递增，在[9,) 上严格递减，且(3)0,(9)1f f ，故结合图像可知(0,3)a ，(3,9)b ，(9,)c ，并且()()()(0,1)f a f b f c ． …………………4分由()()f a f b 得331log log 1a b ，即33log log 2a b ，因此239ab ．于是9abc c ． …………………8分又0()41f c ， …………………12分 故(9,16)c ．进而9(81,144)abc c ．所以，abc 的取值范围是(81,144)． …………………16分注：对任意的(81,144)r ，取09rc =，则0(9,16)c ∈，从而0()(0,1)f c ∈．过点00(,())c f c 作平行于x 轴的直线l ，则l 与()f x 的图像另有两个交点(,())a f a ，(,())b f b （其中(0,3),(3,9)a b ），满足()()()f a f b f c ，并且9ab ，从而abc r =．10.（本题满分20分）已知实数列123,,,a a a 满足：对任意正整数n ，有(2)1n n n a S a ，其中n S 表示数列的前n 项和．证明：(1) 对任意正整数n ，有n a(2) 对任意正整数n ，有11n n a a ．证明：(1) 约定00S ．由条件知，对任意正整数n ，有221111(2)()()n n n n n n n n n a S a S S S S S S ，从而220n S n S n ，即n S （当0n 时亦成立）． …………………5分显然，1n n n a S S ． …………………10分(2) 仅需考虑1,n n a a 同号的情况．不失一般性，可设1,n n a a 均为正（否则将数列各项同时变为相反数，仍满足条件），则11n n n S S S ，故必有1n n S S ，此时1n n a a从而11n n a a ． …………………20分11.（本题满分20分）在平面直角坐标系xOy 中，设AB 是抛物线24y x 的过点(1,0)F 的弦，AOB 的外接圆交抛物线于点P （不同于点,,O A B ）．若PF 平分APB ，求PF 的所有可能值．解：设222123123,,,,,444y y y A y B y P y，由条件知123,,y y y 两两不等且非零． 设直线AB 的方程为1x ty ，与抛物线方程联立可得2440y ty ，故124y y ． ① 注意到AOB 的外接圆过点O ，可设该圆的方程为220x y dx ey ，与24y x 联立得，4210164y d y ey ．该四次方程有123,,,0y y y y 这四个不同的实根，故由韦达定理得12300y y y ，从而312()y y y ．②…………………5分因PF 平分APB ，由角平分线定理知，12PA FA yPB FB y ，结合①、②，有2222312222231212112122222222222321222132()()16(2)44()16(2)()44y y y y y y y y y PA yy PB y y y y y y y y y2222422122122224212112(8)16(416)64192(8)16(416)64192y y y y y y y y y y ， ………………10分 即62226222112122126419264192y y y y y y y y ，故 224224121122()(192)0y y y y y y ． 当2212y y 时，21y y ，故30y ，此时P 与O 重合，与条件不符． 当422411221920y y y y 时，注意到①，有22221212()192()208y y y y ． …………………15分因22121282y y y y ，故满足①以及2212y y 的实数12,y y 存在，对应可得满足条件的点,A B ．此时，结合①、②知222231212()4411444y y y y y PF ．…………………20分2018年全国高中数学联合竞赛加试（A 卷）参考答案及评分标准说明：1． 评阅试卷时，请严格按照本评分标准的评分档次给分．2． 如果考生的解答方法和本解答不同，只要思路合理、步骤正确，在评卷时可参考本评分标准适当划分档次评分，10分为一个档次，不得增加其他中间档次．一、（本题满分40分）设n 是正整数，1212,,,,,,,,,n n a a a b b b A B 均为正实数，满足,,1,2,,i i i a b a A i n ≤≤= ，且1212n n b b b Ba a a A≤ ． 证明：1212(1)(1)(1)1(1)(1)(1)1n n b b b B a a a A ++++≤++++ ．证明：由条件知，1,1,2,,i i i b k i n a =≥= ．记BK A=，则1212n n b b b B a a a A ≤ 化为12n k k k K ≤ ．要证明11111ni i i ik a KA a A =++≤++∏． ① 对1,2,,i n = ，由于1i k ≥及0i a A <≤知，11111111i i i i i i i i i k a k k k A k k a a A A +−−+=−≤−=++++． 结合12n K k k k ≥ 知，为证明①，仅需证明当0,1(1,2,,)i A k i n >≥= 时，有1211111ni n i k A k k k A A A =++≤++∏． ②…………………20分对n 进行归纳．当1n =时，结论显然成立． 当2n =时，由120,,1A k k >≥可知1212122111(1)(1)0111(1)k A k A k k A A k k A A A A +++−−⋅−=−≤++++， ③ 因此2n =时结论成立． …………………30分设n m =时结论成立，则当1n m =+时，利用归纳假设知，11121111111111111m m i i m m m i i k A k A k A k k k A k A A A A A A +++==+++++ =⋅≤⋅ +++++∏∏ 12111m k k k A A ++≤+ ，最后一步是在③中用121,m m k k k k + （注意1211,1m m k k k k +≥≥ ）分别代替12,k k ． 从而1n m =+时结论成立．由数学归纳法可知，②对所有正整数n 成立，故命题得证．…………………40分二、（本题满分40分）如图，ABC 为锐角三角形，AB AC ，M 为BC 边的中点，点D 和E 分别为ABC 的外接圆 BAC和 BC 的中点，F 为ABC 的内切圆在AB 边上的切点，G 为AE 与BC 的交点，N 在线段EF 上，满足NB AB ． 证明：若BN EM ，则DF FG ．（答题时请将图画在答卷纸上）证明：由条件知，DE 为ABC 外接圆的直径，DE BC 于M ，AE AD ． 记I 为ABC 的内心，则I 在AE 上，IF AB ． 由NB AB 可知(180)90NBE ABE ABN ADE90ADE MEI ．① …………………10分又根据内心的性质，有EBI EBC CBI EAC ABI EAB ABI EIB ， 从而BE EI ．结合BN EM 及①知，NBE MEI ≌ ． …………………20分于是90180EMI BNE BFE EFI ，故,,,E F I M 四点共圆．进而可知9090AFM IFM IEM AGM ，从而,,,A F G M 四点共圆． …………………30分 再由90DAG DMG 知，,,,A G M D 四点共圆，所以,,,,A F G M D 五点共圆．从而90DFG DAG ，即DF FG ． …………………40分三、（本题满分50分）设,,n k m 是正整数，满足2k ≥，且21k n m n k−≤<． 设A 是{1,2,,}m 的n 元子集．证明：区间0,1n k−中的每个整数均可表示为a a ′−，其中,a a A ′∈．证明：用反证法．假设存在整数0,1n x k∈ −不可表示为a a ′−，,a a A ′∈．作带余除法m xq r =+，其中0r x ≤<．将1,2,,m 按模x 的同余类划分成x 个公差为x 的等差数列，其中r 个等差数列有1q +项，x r −个等差数列有q 项．由于A 中没有两数之差为x ，故A 不能包含以x 为公差的等差数列的相邻两项．从而1,2,12()22,2|,2q x q q q n A r x r q x r q + ⋅ + =≤+−= ⋅+ ① 这里α 表示不小于α的最小整数． …………………20分由条件，我们有()2121k kn m xq r k k >+−−． ②又0,1n x k ∈ −，故(1)n k x >−． ③情形一：q 是奇数．则由①知，12q n x +≤⋅． ④ 结合②，④可知，1()22121q k kx n xq r xq k k +⋅≥>+≥−−，从而21q k <−．再由q 是奇数可知，23q k ≤−，于是1(1)2q n x k x +≤⋅≤−，与③矛盾．情形二：q 是偶数．则由①知，2qn x r ≤⋅+． ⑤结合②，⑤可知，()221q k x r n xq r k ⋅+≥>+−，从而1(1)2(21)2121xq k k xr k k k −−<<−−−，故2(1)q k <−．再由q 是偶数可知，24q k ≤−，于是(2)(1)2qn x r k x r k x ≤⋅+≤−+<−，与③矛盾．综上可知，反证法假设不成立，结论获证． …………………50分四、（本题满分50分） 数列{}n a 定义如下：1a 是任意正整数， 对整数1n ≥， 1n a +是与1ni i a =∑互素，且不等于1,,n a a 的最小正整数．证明：每个正整数均在数列{}n a 中出现．证明：显然11a =或21a =．下面考虑整数1m >，设m 有k 个不同素因子，我们对k 归纳证明m 在{}n a 中出现．记1n n S a a =++，1n ≥．1k =时，m 是素数方幂，设m p α=，其中0α>，p 是素数．假设m 不在{}n a 中出现．由于{}n a 各项互不相同，因此存在正整数N ，当n N ≥时，都有n a p α>．若对某个n N ≥，n p S ，那么p α与n S 互素，又1,,n a a 中无一项是p α，故由数列定义知1n a p α+≤，但是1n a p α+>，矛盾！因此对每个n N ≥，都有|n p S ．但由1|n p S +及|n p S 知1|n p a +，从而1n a +与n S 不互素，这与1n a +的定义矛盾． …………………10分假设2k ≥，且结论对1k −成立．设m 的标准分解为1212k km p p p ααα=．假设m 不在{}n a 中出现，于是存在正整数N ′，当n N ′≥时，都有n a m >．取充分大的正整数11,,k ββ−，使得11111max k k n n N M p p a ββ−−′≤≤=> ．我们证明，对n N ′≥，有1n a M +≠． …………………20分对任意n N ′≥，若n S 与12k p p p 互素，则m 与n S 互素，又m 在1,,n a a 中均未出现，而1n a m +>，这与数列的定义矛盾．因此我们推出：对任意n N ′≥，n S 与12k p p p 不互素．()∗情形1．若存在(11)i i k ≤≤−，使得|i n p S ，因1(,)1n n a S +=，故1i n p a +，从而1n a M +≠（因|i p M ）． …………………30分 情形2．若对每个(11)i i k ≤≤−，均有i n p S ，则由()∗知必有|k n p S ．于是1k n p a + ，进而1k n n p S a ++，即1k n p S +．故由()∗知，存在00(11)i i k ≤≤−，使得01|i n p S +，再由11n n n S S a ++=+及前面的假设(11)i n p S i k ≤≤−，可知01i n p a +，故1n a M +≠． …………………40分因此对1n N ′≥+，均有n a M ≠，而1max n i N M a ′≤≤>，故M 不在{}n a 中出现，这与归纳假设矛盾．因此，若m 有k 个不同素因子，则m 一定在{}n a 中出现．由数学归纳法知，所有正整数均在{}n a 中出现． …………………50分。
2018年全国高中数学联合竞赛加试参考答案(A卷)
4
(∗) 情形 1．若存在 i (1 ≤ i ≤ k − 1) ，使得 pi | S n ，因 (an+1 , S n ) = 1 ，故 pi an+1 ，从而 ． …………………30 分 an +1 ≠ M （因 pi | M ） 情形 2．若对每个 i (1 ≤ i ≤ k − 1) ，均有 pi S n ，则由 (∗) 知必有 pk | S n ．于是 pk an +1 ，进而 pk S n + an+1 ，即 pk S n+1 ．故由 (∗) 知，存在 i0 (1 ≤ i0 ≤ k − 1) ，使得 pi0 | S n+1 ，再由 S n+ = S n + an+1 及前面的假设 pi S n (1 ≤ i ≤ k − 1) ，可知 pi0 an+1 ，故 1
…………………40 分
1≤i ≤ N ′
an +1 ≠ M ．
因此对 n ≥ N ′ + 1 ， 均有 an ≠ M ， 而 M > max an ， 故 M 不在 {an } 中出现， 这与 归纳假设矛盾．因此，若 m 有 k 个不同素因子，则 m 一定在 {an } 中出现． 由数学归纳法知，所有正整数均在 {an } 中出现． …………………50 分
αk α1 α 2 且结论对 k − 1 成立． 设 m 的标准分解为 m = p1 ． 假设 m 假设 k ≥ 2 ， p2 pk 不在 {an } 中出现，于是存在正整数 N ′ ，当 n ≥ N ′ 时，都有 an > m ．取充分大的正 整数 β1 ,, β k −1 ，使得
k −1 = M p1β1 pkβ− 1 > max an ．
-年amc 10a和b竞赛真题及答案(英文版)演示教学
sink下沉sank / sunksunk / sunken
give给gavegiven
hit打hithit
weave编织wovewoven不规则动词表
2013年AMCபைடு நூலகம்10A竞赛真题及答案（英文版）
2.以d结尾的词，把d变成t。如:build—built, lend—lent, send—sent, spend—spent
2010年AMC 10A竞赛真题及答案（英文版）
2010年AMC 10B竞赛真题及答案（英文版）
2011年AMC 10A竞赛真题及答案（英文版）
2011年AMC 10B竞赛真题及答案（英文版）
Problem 4
2012年AMC 10A竞赛真题及答案（英文版）
go去wentgone
2012年AMC 10B竞赛真题及答案（英文版）
think思考thoughtthought
hit打hithit
hold拿住heldheldlay放置laidlaid
find找出foundfoundmistake误认mistookmistaken
read读readread
misunderstand误会misunderstoodmisunderstood
take拿tooktakengrow成长grewgrown
动词原形中文意思过去式过去分词
spell拼写spelt / spelledspelt / spelled
2010-2015年AMC 10A和B竞赛真题及答案（英文版）
你好，目前，只分享了2010-2013年AMC 10A和B竞赛真题及答案（英文版），2014-2015年的暂时还没有上传，等采纳后再私信我吧！需要的话，我可以将Word文档中的原图文件一同发给你，求采纳，毕竟我花了差不多一个下午才整理完，谢谢！(@_@)