2008 AMC 12B Problems(答案)

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2008 AMC 12B Problems(答案)

2008 AMC 12B ProblemsProblem 1A basketball player made baskets during a game. Each basket was worth either or points. How many different numbers could represent the total points scored by the player?SolutionIf the basketball player makes three-point shots and two-point shots, he scorespoints. Clearly every value of yields a different number of total points. Since he can make any number of three-point shots between and inclusive, the number of different point totals is .Problem 2A block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?SolutionAfter reversing the numbers on the second and fourth rows, the block will look like this:The difference between the two diagonal sums is:.Problem 3A semipro baseball league has teams with players each. League rules state that a player must be paid at least dollars, and that the total of all players' salaries for each team cannot exceed dollars. What is the maximum possiblle salary, in dollars, for a single player?SolutionWe want to find the maximum any player could make, so assume that everyone else makes the minimum possible and that the combined salaries total the maximum ofThe maximum any player could make is dollars (answer choice C)Problem 4On circle , points and are on the same side of diameter , , and. What is the ratio of the area of the smaller sector to the area of the circle?Solution.Since a circle has , the desired ratio is .Problem 5A class collects dollars to buy flowers for a classmate who is in the hospital. Roses cost dollars each, and carnations cost dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly dollars?SolutionThe class could send just carnations (25 of them). They could also send 22 carnations and 2 roses, 19 carnations and 4 roses, and so on, down to 1 carnation and 16 roses. There are 9 total possibilities (from 0 to 16 roses, incrementing by 2 at each step), which is answer choice C.Problem 6Postman Pete has a pedometer to count his steps. The pedometer records up to steps, then flips over to on the next step. Pete plans to determine his mileage for a year. On January Pete sets the pedometer to . During the year, the pedometer flips from to forty-four times. On December the pedometer reads . Pete takes steps per mile. Which of the following is closest to the number of miles Pete walked during the year?SolutionEvery time the pedometer flips, Pete has walked steps. Therefore, Pete has walked a total ofsteps, which is miles, which is closest to answer choice A.Problem 7For real numbers and , define . What is ?SolutionProblem 8Points and lie on . The length of is times the length of , and the length of is times the length of . The length of is what fraction of the length of ?SolutionSince and , .Since and , .Thus, .Problem 9Points and are on a circle of radius and . Point is the midpoint of the minor arc . What is the length of the line segment ?SolutionTrig Solution:Let be the angle that subtends the arc AB. By the law of cosines,The half-angle formula says that, which is answer choice A.Other SolutionDefine D as the midpoint of AB, and R the center of the circle. R, C, and D are collinear, and since D is the midpoint of AB, , and so . Since ,, and soProblem 10Bricklayer Brenda would take hours to build a chimney alone, and bricklayer Brandon would take hours to build it alone. When they work together they talk a lot, and their combined output is decreased by bricks per hour. Working together, they build the chimney in hours. How many bricks are in the chimney?SolutionLet be the number of bricks in the house.Without talking, Brenda and Brandon lay and bricks per hour respectively, so together they lay per hour together.Since they finish the chimney in hours, . Thus, .Problem 11A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?SolutionIn a cone, radius and height each vary inversely with increasing height (i.e. the radius of the cone formed by cutting off the mountain at feet is half that of the original mountain). Therefore, volume varies as the inverse cube of increasing height (expressed as a percentage of the total heightof cone):Plugging in our given condition,, answer choice A.Problem 12For each positive integer , the mean of the first terms of a sequence is . What is the th term of the sequence?SolutionLetting be the nth partial sum of the sequence:The only possible sequence with this result is the sequence of odd integers.Problem 13Vertex of equilateral is in the interior of unit square . Let be the region consistingof all points inside and outside whose distance from is between and . What is the area of ?Problem 14A circle has a radius of and a circumference of . What is ?SolutionLet be the circumference of the circle, and let be the radius of the circle.Using log properties, and .Since , .Problem 15 (no solution)On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let be the region formed by the union of the square and all the triangles, and be the smallest convex polygon that contains . What is the area of the region that is inside but outside ?Baidu查的答案答案是A，其实画个图就清楚了，边长为1的正方形（unit square) 连同周边12个正三角形组成一个新的边长为2的正方形，要使包在正方形外面的八边形面积最小，只有A是正确的，BCEDE 的话面积都比1/4的时候大。

2008 AMC 12A Problems(答案)

2008 AMC 12A ProblemsProblem 1A bakery owner turns on his doughnut machine at . At themachine has completed one third of the day's job. At what time will the doughnut machine complete the job?SolutionThe machine completes one-third of the job in hours. Thus,the entire job is completed in hours.Since the machine was started at , the job will be finished hours later, at. The answer is .Problem 2What is the reciprocal of ?Solution.Problem 3Suppose that of bananas are worth as much as oranges. How many orangesare worth as much as of bananas?SolutionIf , then.Problem 4Which of the following is equal to the productSolutionSolution 1.Solution 2Notice that everything cancels out except for in the numerator and in thedenominator.Thus, the product is , and the answer is .Problem 5Suppose that is an integer. Which of the following statements must be true about ?SolutionFor to be an integer, mustbe even, but not necessarily divisible by . Thus, the answer is .Problem 6Heather compares the price of a new computer at two different stores. Storeoffers off the sticker price followed by a rebate, and store offers offthe same sticker price with no rebate. Heather saves by buying the computer at store instead of store . What is the sticker price of the computer, in dollars?SolutionSolution 1Let the sticker price be .The price of the computer is at store , and at store .Heather saves at store , so .Solving, we find , and the thus answer is .Solution 2The in store is better than the additional off at store .Thus the off is equal to , and therefore the sticker price is .Problem 7While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing toward the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?SolutionIt will take of an hour orminutes to get to shore.Since only gallons of water can enter the boat, only net gallons can enterthe boat per minute. Sincegallons of water enter the boat each minute, LeRoy must bailgallons per minute.Problem 8What is the volume of a cube whose surface area is twice that of a cube with volume 1?SolutionA cube with volume has a side of lengthand thus a surface area of.A cube whose surface area is has a side of length and avolume of .Problem 9Older television screens have an aspect ratio of . That is, the ratio of the widthto the height is . The aspect ratio of many movies is not , so they aresometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of and is shown on an older television screen with a -inchdiagonal. What is the height, in inches, of each darkened strip?SolutionLet the width and height of the screen be and respectively, and let the widthand height of the movie be and respectively.By the Pythagorean Theorem, the diagonal is . So.Since the movie and the screen have the same width, .Thus, the height of each strip is .Problem 10Doug can paint a room in hours. Dave can paint the same room in hours. Dougand Dave paint the room together and take a one-hour break for lunch. Let be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by ?SolutionDoug can paint of a room per hour, Dave can paint of a room in an hour, and the time they spend working together is .Since rate times time gives output,Problem 11Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the visible numbers have the greatest possiblesum. What is that sum?SolutionTo maximize the sum of the faces that are showing, we can minimize the sum ofthe numbers of the faces that are not showing.The bottom cubes each have a pair of opposite faces that are covered up. Whenthe cube is folded, ; ; and are opposite pairs. Clearlyhas the smallest sum.The top cube has 1 number that is not showing. The smallest number on a face is .So, the minimum sum of the unexposed faces is . Since the sum ofthe numbers on all the cubes is , the maximumpossible sum of visible numbers is .Problem 12A function has domain and range. (The notation denotes.) What are the domain and range, respectively, of the functiondefined by ?Solutionis defined if is defined. Thus the domain is all.Since , . Thusis the range of .Thus the answer is .Problem 13Points and lie on a circle centered at , and . A second circle isinternally tangent to the first and tangent to both and . What is the ratio of the area of the smaller circle to that of the larger circle?SolutionLet be the center of the small circle with radius , and let be the point wherethe small circle is tangent to . Also, let be the point where the small circle istangent to the big circle with radius .Then is a right triangle, and a triangle at that. Therefore,.Since , we have , or, or .Then the ratio of areas will be squared, or .Problem 14What is the area of the region defined by the inequality?Area is invariant under translation, so after translating left and up units, wehave the inequalitywhich forms a diamond centered at the origin and vertices at .Thus the diagonals are of length and . Using the formula , theanswer is .Problem 15Let . What is the units digit of ?Solution.So, . Since is a multiple of four and the unitsdigit of powers of two repeat in cycles of four, .Therefore, . So the units digit is .Problem 16The numbers , , and are the first three terms of anarithmetic sequence, and the term of the sequence is . What is ?Solution 1Let and .The first three terms of the arithmetic sequence are , , and, and the term is .Thus, .Since the first three terms in the sequence are , , and , the th termis .Thus the term is .Solution 2If , , and are in arithmetic progression, then ,, and are in geometric progression. Therefore,Therefore, , , therefore the 12th term in the sequence isProblem 17Let be a sequence determined by the rule if is evenand if is odd. For how many positive integers is it true that is less than each of , , and ?SolutionAll positive integers can be expressed as , , , or , whereis a nonnegative integer.▪ If , then .▪ If , then ,, and.▪If, then.▪ If , then,, and.Since , every positive integerwill satisfy.Since one fourth of the positive integers can be expressed as,where is a nonnegative integer, the answer is.Problem 18Triangle, with sides of length , , and , has one vertex on the positive-axis, one on the positive -axis, and one on the positive -axis. Let be the origin .What is the volume of tetrahedron?SolutionWithout loss of generality, let be on the axis, be on the axis, and be onthe axis, and let have respective lengths of 5, 6, and 7. Letdenote the lengths of segments respectively. Then by thePythagorean Theorem, so ; similarly,and . Since , , and are mutually perpendicular, thetetrahedron's volume is which is answer choice C.Problem 19In the expansion of what isthe coefficient of ?SolutionLet and . We areexpanding .Since there are terms in , there are ways to choose one term fromeach . The product of the selected terms is for some integer between andinclusive. For each , there is one and only one in . Since there isonly one way to choose one term from each to get a product of , there areways to choose one term from each and one term from to get aproduct of . Thus the coefficient of the term is .Problem 20Triangle has , , and . Point is on , andbisects the right angle. The inscribed circles of and have radiiand , respectively. What is ?SolutionBy the Angle Bisector Theorem, By Law of Sines on, Since the areaof a triangle satisfies , where the inradius and the semiperimeter,we have Since and share the altitude (to ),their areas are the ratio of their bases, or The semiperimetersare and . Thus,Problem 21A permutation of is heavy-tailed if. What is the number of heavy-tailed permutations?SolutionThere are total permutations.For every permutation such that , there isexactly one permutation such that . Thus it suffices to count thepermutations such that ., , and are the only combinations of numbers that can satisfy .There are combinations of numbers, possibilities of which side of the equation isand which side is , and possibilities for rearrangingand . Thus, there are permutations such that.Thus, the number of heavy-tailed permutations is .Problem 22A round table has radius . Six rectangular place mats are placed on the table. Eachplace mat has width and length as shown. They are positioned so that each mathas two corners on the edge of the table, these two corners being end points of the same side of length . Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is ?SolutionSolution 1 (trigonometry)Let one of the mats be , and the center be as shown:Since there are mats, is equilateral. So, . Also,.By the Law of Cosines:.Since must be positive, .Solution 2 (without trigonometry)Draw and as in the diagram. Draw the altitude from to and call theintersectionAs proved in the first solution, . That makes atriangle, so andSince is a right triangle,Solving for givesProblem 23The solutions of the equation are the vertices of a convex polygon in the complex plane. What is the area of the polygon?SolutionLooking at the coefficients, we are immediately reminded of the binomial expansionof .Modifying this slightly, we can write the given equation as:We can apply a translation of and a rotation of(both operations preserve area) to simplify the problem:Because the roots of this equation are created by rotating radians successively about the origin, the quadrilateral is a square.We know that half the diagonal length of the square isTherefore, the area of the square isProblem 24Triangle has and . Point is the midpoint of . What isthe largest possible value of ?SolutionLet . Then , and sinceand , we haveWith calculus, taking the derivative and setting equal to zero will give the maximum value of . Otherwise, we can apply AM-GM:Thus, the maximum is at .Problem 25A sequence , , , of points in the coordinate plane satisfiesfor .Suppose that . What is ?SolutionThis sequence can also be expressed using matrix multiplication as follows:.Thus, is formed by rotating counter-clockwise about the originbyand dilating the point's position with respect to the origin by a factor of .So, starting with and performing the above operationstimes inreverse yields.Rotating clockwise by yields . A dilation by a factor ofyields the point .Therefore, .。

2013 AMC 12B Problems

On a particular January day, the high temperature in Lincoln, Nebraska, was degrees higher than the low temperature, and the average of the high and low temperatures was . In degrees, what was the low temperature in Lincoln that day?SolutionMr. Green measures his rectangular garden by walking two of the sides and finds that it is stepsby steps. Each of Mr. Green’s steps is feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?SolutionWhen counting from to , is the number counted. When counting backwardsfrom to , is the number counted. What is ?SolutionRay's car averages miles per gallon of gasoline, and Tom's car averages miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?SolutionThe average age of fifth-graders is . The average age of of their parents is . What is the average age of all of these parents and fifth-graders?SolutionReal numbers and satisfy the equation . What is ?SolutionJo and Blair take turns counting from to one more than the last number said by the other person. Jo starts by saying , so Blair follows by saying . Jo then says , and so on. What is the number said?SolutionLine has equation and goes through . Line has equation and meets line at point . Line has positive slope, goes through point , and meets at point . The area of is . What is the slope of ?SolutionWhat is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides ?SolutionAlex has red tokens and blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokensand receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?SolutionTwo bees start at the same spot and fly at the same rate in the following directions. Bee travels foot north, then foot east, then foot upwards, and then continues to repeat this pattern. Beetravels foot south, then foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly feet away from each other?east, westnorth, southnorth, westup, southup, westSolutionCities , , , , and are connected by roads , , , , , , and . How many different routes are there from to that use each road exactly once? (Such a route will necessarilyvisit some cities more than once.)SolutionThe internal angles of quadrilateral form an arithmetic progression.Triangles and are similar with and . Moreover, theangles in each of these two triangles also form an arithmetic progression. In degrees, what is the largestpossible sum of the two largest angles of ?SolutionTwo non-decreasing sequences of nonnegative integers have different first terms. Each sequence has theproperty that each term beginning with the third is the sum of the previous two terms, and the seventhterm of each sequence is . What is the smallest possible value of ?SolutionThe number is expressed in the form,where and are positive integers and is as small as possible. What is ?SolutionLet be an equiangular convex pentagon of perimeter . The pairwise intersections of the linesthat extend the sides of the pentagon determine a five-pointed star polygon. Let be the perimeter of thisstar. What is the difference between the maximum and the minimum possible values of ?SolutionLet and be real numbers such thatWhat is the difference between the maximum and minimum possible values of ?SolutionBarbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove or coins, unless only one coin remains, in which case she loses her turn. What it is Jenna’s turn, she must remove or coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with coins and when the game starts with coins?Barbara will win with coins and Jenna will win with coins.Jenna will win with coins, and whoever goes first will win with coins.Barbara will win with coins, and whoever goes second will win with coins.Jenna will win with coins, and Barbara will win with coins.Whoever goes first will win with coins, and whoever goes second will win with coins. SolutionIn triangle , , , and . Distinct points , , and lie onsegments , , and , respectively, such that , , and . The length of segment can be written as , where and are relatively prime positive integers. What is ?SolutionFor ,points and are the vertices of a trapezoid. What is ?SolutionConsider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form with a and b integers suchthat and . No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?SolutionLet and be integers. Suppose that the product of the solutions for of the equationis the smallest possible integer. What is ?SolutionBernardo chooses a three-digit positive integer and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer . For example, if , Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum . For how many choices of are the two rightmost digits of , in order, the same as those of ?SolutionLet be a triangle where is the midpoint of , and is the angle bisectorof with on . Let be the intersection of the median and the bisector . In addition is equilateral with . What is ?SolutionLet be the set of polynomials of the formwhere are integers and has distinct roots of the form with and integers. How many polynomials are in ?Solution。

amc12真题及答案

amc12真题及答案Problem 1What is the value of ?SolutionProblem 2For what value of does ?SolutionProblem 3The remainder can be defined for all real numbers and with bywhere denotes the greatest integer less than or equal to . What is the value of ?SolutionProblem 4The mean, median, and mode of the data values are all equal to . What is the value of ?SolutionProblem 5Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, ). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?SolutionProblem 6A triangular array of coins has coin in the first row, coins in the second row, coins in the third row, and so on up to coins in the th row. What is the sum of the digits of ?SolutionProblem 7Which of these describes the graph of ?SolutionProblem 8What is the area of the shaded region of the given rectangle?SolutionProblem 9The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is , where and are positive integers. What is ?SolutionProblem 10Five friends sat in a movie theater in a row containing seats, numbered to from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seatsto the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?SolutionProblem 11Each of the students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. Thereare students who cannot sing, students who cannot dance, and students who cannot act. How many students have two ofthese talents?SolutionProblem 12In , , , and . Point lies on , and bisects . Point lies on ,and bisects . The bisectors intersect at . What is the ratio : ?SolutionProblem 13Let be a positive multiple of . One red ball and green balls are arranged in a line in random order. Let be the probability that at least of the green balls are on the same side of the red ball. Observe that andthat approaches as grows large. What is the sum of the digits of the least value of such that ?SolutionProblem 14Each vertex of a cube is to be labeled with an integer from through , with each integer being used once, in such a way that the sum of the four numbers on the verticesof a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?SolutionProblem 15Circles with centers and , having radii and , respectively, lie on the same side of line and are tangent to at and , respectively, with between and . The circle with center is externally tangent to each of the other two circles. What is the area of triangle ?SolutionProblem 16The graphs of and are plotted on the same set of axes. How many points in the plane with positive -coordinates lie on two or more of the graphs?SolutionProblem 17Let be a square. Let and be the centers, respectively, of equilateral triangles with bases and each exterior to the square. What is the ratio of the area of square to the area of square ?SolutionProblem 18For some positive integer the number has positive integer divisors, including and the number How many positive integer divisors does thenumber have?SolutionProblem 19Jerry starts at on the real number line. He tosses a fair coin times. When he gets heads, he moves unit in the positive direction; when he gets tails, he moves unit in the negative direction. The probability that he reaches at some time during this process is where and are relatively prime positive integers. What is (For example, he succeeds if his sequence of tosses is ) SolutionProblem 20A binary operation has the properties that and that for all nonzero real numbers and (Here the dot represents the usual multiplication operation.) The solution to the equation can be written as where and are relatively prime positive integers. What isSolutionProblem 21A quadrilateral is inscribed in a circle of radius Three of the sides of this quadrilateral have length What is the length of its fourth side?SolutionProblem 22How many ordered triples of positive integers satisfy and ?SolutionProblem 23Three numbers in the interval are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?SolutionProblem 24There is a smallest positive real number such that there exists a positive real number such that all the roots of the polynomial are real. In fact, for this value of the value of is unique. What is the value ofSolutionProblem 25Let be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with digits. Every time Bernardo writes a number, Silvia erases the last digits of it. Bernardo then writes the next perfect square, Silvia erases the last digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let be the smallest positive integer not written on the board. For example, if , then the numbers that Bernardo writes are , and the numbersshowing on the board after Silvia erases are and , and thus . What is the sum of the digits of ?2016 AMC 12A Answer Key1 B2 C3 B4 D5 E6 D7 D8 D9 E10 B11 E12 C13 A14 C15 D16 D17 B18 D19 B20 A21 E22 A23 C24 B25 E。

AMC Junior 2008澳大利亚数学AMC-C：7-8年级英语历年真题

A u s t r A l i A n M At h e M At i c s c o M p e t i t i o na n a c t i v i t y o f t h e a u s t r a l i a n m a t h e m a t i c s t r u s tT H U R S D AY 31 J U LY 2008JUNIOR DIVISION COMPETITION PAPERA U S T R A L I A N S C H O O LY E A R S 7 A N D 8T I M E A L L O W E D : 75 M I N U T ESJunior DivisionQuestions 1to 10,3marks each1.The value of 2008+8002is (A)1010(B)4004(C)10008(D)8910(E)100102.Which of the following numbers has the largest value?(A)2.15(B)2.2(C)2.08(D)2.1(E)2.1853.The perimeter of the ﬁgure,in centimetres,is (A)8(B)10(C)12(D)16(E)20......................................................4cm2cm 4cm2cm4.One half of 19912is (A)9512(B)9534(C)9914(D)9912(E)99345.The value of x is(A)135(B)95(C)35(D)55(E)45...................................................................................................................135◦x ◦6.The value of 200×8200÷8is(A)1(B)8(C)16(D)64(E)2007.How many squares of any size are there in the diagram?(A)9(B)11(C)12(D)14(E)161111111122J28.A train left Fassifern at8:58am and arrived at Broadmeadow at9:34am on thesame day.The time taken,in minutes,was(A)82(B)22(C)36(D)38(E)789.The digits5,6,7,8and9can be arranged to form evenﬁve-digit numbers.Thetens digit in the largest of these numbers is(A)5(B)6(C)7(D)8(E)910.P QRS is a square and points E and F are outside the square so that P QE andQRF are equilateral triangles.The size of EQF,in degrees,is(A)60(B)90(C)120(D)150(E)180Questions11to20,4marks each11.A rectangle has an area of72square centimetres and the length is twice the width.The perimeter,in centimetres,of the rectangle is(A)34(B)36(C)42(D)48(E)5412.Marbles of three diﬀerent colours are in a tin and 25of the marbles are red,13aregreen and the remaining12are yellow.The number of marbles in the tin is(A)30(B)45(C)54(D)60(E)9013.In the diagram,triangles P QR and LMN areboth equilateral and QSM=20◦.What is the value of x?(A)70(B)80(C)90(D)100(E)110....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................LNM PQRS20◦x◦J314.At half-time in a soccer match between Newcastle and Melbourne,the score wasNewcastle1,Melbourne0.Three goals were scored in the second half.Which of the following could not be the result of the match?(A)The match was drawn(B)Newcastle won by2goals(C)Melbourne won by2goals(D)Newcastle won by1goal(E)Newcastle won by4goals15.In how many ways can12be written as the sum of two or more diﬀerent positivewhole numbers?(Changing the order of addition does not count as a diﬀerent way.)(A)12(B)13(C)14(D)15(E)1616.How many diﬀerent positive numbers are equal to the product of two odd one-digitnumbers?(A)25(B)15(C)14(D)13(E)1117.The perimeter of this rectangular paddock is700m.It is subdivided into sixidentical paddocks as shown.The perimeter,in metres,of each of the six smaller paddocks is(A)11613(B)300(C)200(D)150(E)60018.The student lockers at Euler High School are to be numbered consecutively from1to500using plastic digits which cost5cents each.The total cost of all the digits will be(A)$25(B)$63.65(C)$69.50(D)$69.60(E)$85J419.In the grid below,the squares are to beﬁlled with the numbers1,2,3and4sothat they appear once only in each row,each column and each diagonal.123XYThe largest possible value of X+Y is(A)4(B)5(C)6(D)7(E)820.The average of one group of numbers is4.A second group contains twice asmany numbers and has an average of10.The average of both groups of numbers combined is(A)5(B)6(C)7(D)8(E)9Questions21to25,5marks each21.A cube with edge length2metres is cut up into cubes each with edge length5centimetres.If all these cubes were stacked one on top of the other to form a tower, the height of the tower would be(A)32km(B)160m(C)1600m(D)3.2km(E)320m22.A number is less than2008.It is odd,it leaves a remainder of2when divided by3and a remainder of4when divided by5.What is the sum of the digits of the largest such number?(A)26(B)25(C)24(D)23(E)2223.Farmer Taylor of Burra has two tanks.Water from the roof of his farmhouse iscollected in a100kL tank and water from the roof of his barn is collected in a 25kL tank.The collecting area of his farmhouse roof is200square metres while that of his barn is80square metres.Currently,there are35kL in the farmhouse tank and13kL in the barn tank.Rain is forecast and he wants to collect as much water as possible.He should:(A)empty the barn tank into the farmhouse tank(B)ﬁll the barn tank from the farmhouse tank(C)pump10kL from the farmhouse tank into the barn tank(D)pump10kL from the barn tank into the farmhouse tank(E)do nothingJ 524.A ﬁshtank with base 100cm by 200cm and depth 100cm contains water to a depthof 50cm.A solid metal rectangular prism with dimensions 80cm by 100cm by 60cm is then submerged in the tank with an 80cm by 100cm face on the bottom..................................................................................................................6.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................100100200501006080The depth of water,in centimetres,above the prism is then (A)12(B)14(C)16(D)18(E)2025.A strip of paper is folded in a line at an angle x ◦to the sides and then foldedunderneath forming an angle of 20◦as shown...............x ◦=⇒........................................................................................................................................................................................................................................................................................=⇒...............................................................................................................................................................................................................20◦The value of x is (A)60(B)65(C)70(D)75(E)80For questions 26to 30,shade the answer as an integer from 0to 999inthe space provided on the answer sheet.Question 26is 6marks,question 27is 7marks,question 28is 8marks,question 29is 9marks and question 30is 10marks.26.A two-digit number ab and its reversal ba are both prime.For example,13and 31are both prime.What is the largest possible sum of these two numbers ab and ba ?J627.Given a regular heptagon(7-sided polygon),how many obtuse-angled triangles arethere,where the vertices of each triangle are vertices of the heptagon?28.A rectangular prism6cm by3cm by3cm is made up by stacking1cm by1cmby1cm cubes.How many rectangular prisms,including cubes,are there whose vertices are vertices of the cubes,and whose edges are parallel to the edges of the original rectangular prism?(Rectangular prisms with the same dimensions but in diﬀerent positions are diﬀerent.)29.Let us call a sum of integers cool if theﬁrst and last terms are1and each termdiﬀers from its neighbours by at most1.For example,the sum1+2+3+4+3+ 2+3+3+3+2+3+3+2+1is cool.How many terms does it take to write2008as a cool sum if we use no more terms than necessary?30.A monument has been constructed from identical stone cubes.The views fromabove,the front f and the side s are shown.sfabove view front view side view What is the largest number of stones in the monument consistent with these views?***Junior 2008 Answers Question Answer 1E2B3D4E5E6D7D8C9A10D11B12B13B14D15C16C17B18D19C20D21D22A23D24B25E26176272128756298930106。

AMC2008

AMC2008年真题Problem 1Susan had dollars to spend at the carnival. She spent dollars on food and twice as much on rides. How many dollars did she have left to spend?Problem 2The ten-letter code represents the ten digits , in order. What4-digit number is represented by the code word ?Problem 3If February is a month that contains Friday the , what day of the week is February 1?Problem 4In the figure, the outer equilateral triangle has area , the inner equilateral triangle has area , and the three trapezoids are congruent. What is the area of one of the trapezoids?Problem 5Barney Schwinn notices that the odometer on his bicycle reads , a palindrome, becauseit reads the same forward and backward. After riding more hours that day and the next,he notices that the odometer shows another palindrome, . What was his average speed in miles per hour?In the figure, what is the ratio of the area of the gray squares to the area of the white squares?Problem 7If , what is ?Problem 8Candy sales from the Boosters Club from January through April are shown. What were theaverage sales per month in dollars?Problem 9In Tycoon Tammy invested dollars for two years. During the the first year her investment suffered a loss, but during the second year the remaining investmentshowed a gain. Over the two-year period, what was the change in Tammy's investment?The average age of the people in Room A is . The average age of the people in RoomB is . If the two groups are combined, what is the average age of all the people?Problem 11Each of the students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and students have a cat. How manystudents have both a dog and a cat?Problem 12A ball is dropped from a height of meters. On its first bounce it rises to a height of meters.It keeps falling and bouncing to of the height it reached in the previous bounce. On which bounce will it not rise to a height of meters?Problem 13Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than pounds or more than pounds. So the boxes are weighed in pairs in every possible way. The results are, and pounds. What is the combined weight in pounds of the three boxes?Three , three , and three are placed in the nine spaces so that each row andcolumn contain one of each letter. If is placed in the upper left corner, how manyarrangements are possible?Problem 15In Theresa's first basketball games, she scored and points. In her ninthgame, she scored fewer than points and her points-per-game average for the nine gameswas an integer. Similarly in her tenth game, she scored fewer than points and herpoints-per-game average for the games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?Problem 16A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?Problem 18Two circles that share the same center have radii meters and meters. An aardvark runs along the path shown, starting at and ending at . How many meters does theaardvark run?Problem 19Eight points are spaced around at intervals of one unit around a square, as shown.Two of the points are chosen at random. What is the probability that the two points are oneunit apart?The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?Problem 21Jerry cuts a wedge from a -cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?Problem 22For how many positive integer values of are both and three-digit whole numbers?In square , and . What is the ratio of the area ofto the area of square ?Problem 24Ten tiles numbered through are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?Problem 25Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Approximately what percent of the design is black?。

2002AMC12B解答

A A/2 = minutes to mow hers, 2R 4R A/3 A = minutes to mow his. R 3R
A A < , Beth will ﬁnish ﬁrst. 4R 3R
18. (C) The area of the rectangular region is 2. Hence the probability that P is closer to (0, 0) than it is to (3, 1) is half the area of the trapezoid bounded by the lines y = 1, the x- and y - axes, and the perpendicular bisector of the segment joining (0, 0) and (3, 1). The perpendicular bisector goes through the point (3/2, 1/2), which is the center of the square whose vertices are (1, 0), (2, 0), (2, 1), and Hence, the line cuts the square into two quadrilaterals of equal area 1/2. Thus the area of the trapezoid is 3/2 and the probability is 3/4.
1 41 1 1 1 1 + + + = + 2 3 7 n 42 n is an integer precisely when it is equal to 1. This implies that n = 42, so the answer is (E).

2011 AMC 12B Problems and Solution

2011 AMC 12B Problems and Solution Problem 1What isSolutionAdd up the numbers in each fraction to get , which equals . Doingthe subtraction yieldsProblem 2Josanna's test scores to date are , , , , and . Her goal is to raise hertest average at least points with her next test. What is the minimum test score shewould need to accomplish this goal?SolutionTake the average of her current test scores, which isThis m eans that she wants her test average after the sixth test to be Let be thescore that Josanna receives on her sixth test. Thus, our equation isProblem 3LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paiddollars and Bernardo had paid dollars, where . How m any dollars mustLeRoy give to Bernardo so that they share the costs equally?SolutionThe total am ount of m oney that was spent during the trip was So eachperson should pay if they were to share the costs equally. Because LeRoyhas already paid dollars of his part, he still has to payProblem 4In multiplying two positive integers and , Ron reversed the digits of the two-digitnumber . His erroneous product was 161. What is the correct value of the product of and ?SolutionTaking the prime factorization of reveals that it is equal to Therefore, theonly ways to represent as a product of two positive integers is andBecause neither nor is a two-digit number, we know that and areand Because is a two-digit number, we know that a, with its two digits reversed,gives Therefore, and Multiplying our two correct values of andyieldsProblem 5Let be the second sm allest positive integer that is divisible by every positiveinteger less than . What is the sum of the digits of ?Solutionmust be divisible by every positive integer less than , or and . Eachnumber that is divisible by each of these is is a multiple of their least commonmultiple. , so each num ber divisible by these is a multiple of . The sm allest m ultiple of is clearly , so the second sm allestmultiple of is . Therefore, the sum of the digits of isProblem 6Two tangents to a circle are drawn from a point . The points of contact anddivide the circle into arcs with lengths in the ratio . What is the degree m easureof ?SolutionIn order to solve this problem, use of the tangent-tangent intersection theorem (Angle of intersection between two tangents dividing a circle into arc length A and arc length B = 1/2 (Arc A° - Arc B°).In order to utilize this theorem, the degree measures of the arcs must be found. First, set A (Arc length A) equal to 3d, and B (Arc length B) equal to 2d.Setting 3d+2d = 360° will find d = 72°, and so therefore Arc length A in degrees will equal 216° and arc length B will equal 144°.Finally, simply plug the two arc lengths into the tangent-tangent intersection theorem, and the answer:1/2 (216°-144°) = 1/2 (72°)Problem 7Let and be two-digit positive integers with m ean . What is the m aximumvalue of the ratio ?SolutionIf and have a m ean of , then and . To m aximize ,we need to m aximize and minimize . Since they are both two-digit positiveintegers, the m aximum of is which gives . cannot be decreasedbecause doing so would increase , so this gives the m aximum value of , which isProblem 8Keiko walks once around a track at exactly the sam e constant speed every day. The sides of the track are straight, and the ends are sem icircles. The track has widthmeters, and it takes her seconds longer to walk around the outside edge of thetrack than around the inside edge. What is Keiko's speed in meters per second?SolutionTo find Keiko's speed, all we need to find is the difference between the distance around the inside edge of the track and the distance around the outside edge of the track, and divide it by the differenc e in tim e it takes her for each distance. We aregiven the difference in tim e, so all we need to find is the difference between the distances.The track is divided into lengths and curves. The lengths of the track will exhibit no difference in distance between the inside and outside edges, so we only need to concern ourselves with the curves.The curves of the track are semicircles, but since there are two of them, we can consider both of the at the sam e time by treating them as a single circle. We need to find the difference in the circum ferences of the inside and outside edges of the circle.The form ula for the circum ference of a circle is where is the radiusof the circle.Let's define the circum ference of the inside circle as and the circum ference of theoutside circle as .If the radius of the inside circle () is , then given the thickness of the track is 6meters, the radius of the outside circle () is .Using this, the difference in the circum ferences is:is the difference between the inside and outside lengths of the track. Divided by the tim e differential, we get:Problem 9Two real numbers are selected independently and at random from the interval. What is the probability that the product of those num bers is greater than zero?SolutionFor the product to be greater than zero, we m ust have either both numbers negative or both positive.Both numbers are negative with a chance.Both numbers are positive with a chance.Therefore, the total probability is and we are done.Problem 10Rectangle has and . Point is chosen on side sothat . What is the degree m easure of ?SolutionSince , hence . Therefore. ThereforeProblem 11A frog located at , with both and integers, makes successive jumps oflength and always lands on points with integer coordinates. Suppose that the frogstarts at and ends at . What is the sm allest possible number of jumps the frog m akes?SolutionSince the frog always jumps in length and lands on a lattice point, the sum of itscoordinates m ust change either by (by jumping parallel to the x- or y-axis), or byor (based off the 3-4-5 right triangle).Because either , , or is always the change of the sum of the coordinates, the sum of the coordinates will always change from odd to even or vice versa. Thus, itis impossible for the frog to go from to in an even number of moves.Therefore, the frog cannot reach in two m oves.However, a path is possible in 3 m oves: from to to to .Thus, the answer is .Problem 12A dart board is a regular octagon divided into regions as shown below. Suppose t hat a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?SolutionLet's assum e that the side length of the octagon is . The area of the center squareis just . The triangles are all triangles, with a side length ratio of. The area of each of the identical triangles is , so thetotal area of all of the triangles is also . Now, we must find the area of all of the 4 identical rectangles. One of the side lengths is and the other side length is, so the area of all of the rectangles is . The ratio of the area ofthe square to the area of the octagon is . Cancelling from thefraction, the ratio becom es . Multiplying the numerator and thedenominator each by will cancel out the radical, so the fraction is nowProblem 13Brian writes down four integers whose sum is . The pairwisepositive differences of these num bers are and . What is the sum of thepossible values of ?SolutionAssum e that results in the greatestpairwise difference, and thus it is . This m eans . must be inthe set . The only way for 3 num bers in the set to add up to 9 is if they are. , and then must be the rem aining two numbers which are and. The ordering of must be either or .Case 1Case 2The sum of the two w's isProblem 14A segm ent through the focus of a parabola with vertex is perpendicular toand intersects the parabola in points and . What is ?SolutionName the directrix of the parabola . Define to be the distance between apoint and a line .Now we remember the geom etric definition of a parabola: given any line (calledthe directrix) and any point (called the focus), the parabola corresponding to thegiven directrix and focus is the locus of the points that are equidistant from and. Therefore . Let this distance be . Now note that , so. Therefore . We now use the PythagoreanTheorem on triangle ; . Similarly,. We now use the Law of Cosines:This shows that the answer is .Problem 15How many positive two-digit integers are factors of ?SolutionFrom repeated application of difference of squares:Aplying sum of cubes:A quick check shows is prime. Thus, the only factors to be concerned about are, since multiplying by will make any factor too large.Multiply by or will give a two digit factor; itself will also work. The nextsm allest factor, , gives a three digit number. Thus, there are factors which aremultiples of .Multiply by or will also give a two digit factor, as well as itself. Highernumbers will not work, giving an additional factors.Multiply by or for a two digit factor. There are no mare factors to check, asall factors which include are already counted. Thus, there are an additionalfactors.Multiply by or for a two digit factor. All higher factors have been countedalready, so there are more factors.Thus, the total num ber of factors isProblem 16Rhombus has side length and . Region consists of all pointsinside of the rhombus that are closer to vertex than any of the other threevertices. What is the area of ?SolutionSuppose that is a point in the rhombus and let be the perpendicularbisector of . Then if and only if is on the sam e side of as .The line divides the plane into two half-planes; let be the half-planecontaining . Let us define similarly and . Then is equal to. The region turns out to be an irregular pentagon. We can m ake it easier to find the area of this region by dividing it into four triangles:Since and are equilateral,contains , contains and , and contains . Thenwith and soand has area .Problem 17Let , andfor integers . What is the sum of the digits of ?SolutionProof by induction that :ForAssum e is true for n:Therefore, if it is true for n, then it is true for n+1; since it is also true for n = 1, it is true for all positive integers n., which is the 2011-digitnumber 8888 (8889)The sum of the digits is 8 tim es 2010 plus 9, orProblem 18A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?SolutionWe can use the Pythagorean Theorem to split one of the triangular faces into two30-60-90 triangles with side lengths and .Next, take a cross-section of the pyram id, forming a triangle with the top of the triangle and the midpoints of two opposite sides of the square base.This triangle is isosceles with a base of 1 and two sides of length .The height of this triangle will equal the height of the pyram id. To find this height,split the triangle into two right triangles, with sides and .Sorry, the GeoGebra Applet could not be started. Please m ake sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now)The cube, touching all four triangular faces, will form a similar pyramid which sits ontop of the cube. If the cube has side length , the pyramid has side length .Thus, the height of the cube plus the height of the sm aller pyramid equals the height of the larger pyramid..side length of cube.Problem 19A lattice point in an -coordinate system is any point where both and areintegers. The graph of passes through no lattice point withfor all such that . What is the m aximum possible value of ?SolutionIt is very easy to see that the in the graph does not impact whether it passes through lattice.We need to m ake sure that cannot be in the form of for , otherwise the graph passes through lattice point at . We only need to worryabout very close to , , will be the only case we need to worryabout and we want the minimu m of those, clearly for , the sm allest is, so answer isProblem 20Triangle has , and . The points , andare the midpoints of , and respectively. Let be the intersectionof the circum circles of and . What is ?Solution 1Answer: (C)Let us also consider the circum circle of .Note that if we draw the perpendicular bisector of each side, we will have thecircum center of which is , Also, since .is cyclic, similarly, and are also cyclic. With this, we knowthat the circum circles of , and all intercept at , so is.The question now becom es calculate the sum of distance from each vertices to the circum center.We can do it will coordinate geometry, note that because ofbeing circum center.Let , , ,Then is on the line and also the line with slope and passes through.SoandSolution 2Consider an additional circum circle on . After drawing the diagram, it isnoticed that each triangle has side values: , , . Thus they are congruent, and their respective circum circles are. By inspection, we see that , , andare the circum diameters, and so they are congruent. Therefore, the solution can be found by calculating one of these circum diam eters and multiplying it by a factor of . We can find the circumradius quite easily with the form ula, s.t. and R is the circumradius.Since :After a few algebraic m anipulations:.Problem 21The arithm etic m ean of two distinct positive integers and is a two-digit integer. The geom etric m ean of and is obtained by reversing the digits of the arithm eticmean. What is ?SolutionAnswer: (D)for som e ,.Note that in order for x-y to be integer, has to be for some perfectsquare . Since is at m ost , orIf , , if , . In AMC, we are done. Otherwise,we need to show that is impossible.->, or or and , , respectively.And since , , , but there is no integer solution for , .Problem 22Let be a triangle with sides , and . For , ifand , and are the points of tangency of the incircle of to the sides, and , respectively, then is a triangle with side lengths ,and , if it exists. What is the perim eter of the last triangle in the sequence ?SolutionAnswer: (D)Let , , andThen , andThen , ,Hence:Note that and for , I claim that it is true for all , assum e for induction that it is true for som e , thenFurthermore, the average for the sides is decreased by a factor of 2 each tim e.So is a triangle with side length , ,and the perimeter of such isNow we need to find what fails the triangle inequality. So we need to find the lastsuch thatFor , perimeter isProblem 23A bug travels in the coordinate plane, moving only along the lines that are paralle lto the -axis or -axis. Let and . Consider all possible paths of the bug from to of length at m ost . How m any points with integercoordinates lie on at least one of these paths?SolutionAnswer: (C)If a point satisfy the property that ,then it is in the desire range because is the shortest path fromto , and is the shortest path from toIf , then satisfy the property. there arelattice points here.else let (and for it is symmetrical,,So for , there are lattice points,for , there are lattice points,etc.For , there are lattice points.Hence, there are a total of lattice points. Problem 24Let . What is the minimum perimeter am ong all the -sided polygons in the complex plane whose vertices are preciselythe zeros of ?SolutionAnswer: (B)First of all, we need to find all such thatSo ororNow we have a solution at if we look at them in polar coordinate, further m ore, the 8-gon is symm etric (it is an equilateral octagon) . So we only need to find the side length of one and multiply by .So answer distance from toSide lengthHence, answer is .Problem 25For every and integers with odd, denote by the integer closest to . Forevery odd integer , let be the probability thatfor an integer randomly chosen from the interval . What is theminimum possible value of over the odd integers in the interval?SolutionAnswer:First of all, you have to realize thatifthenSo, we can consider what happen in and it will repeat. Also since range ofis to , it is always a m ultiple of . So we can just consider for.LET be the fractional part functionThis is an AMC exam, so use the given choices wisely. With the given choices, andthe previous explanation, we only need to consider , , , .For , . 3 of the that should consider lands in here.For , , then we needelse for , , then we needFor ,So, for the condition to be true, . ( , no worry for therounding to be ), so this is always true.For , , so we want , orFor k = 67,For k = 69,etc.We can clearly see that for this case, has the m inimu m, which is .Also, .So for AMC purpose, answer is (D).Now, let's say we are not given any answer, we need to consider .I claim thatIf got round down, then all satisfy the condition along withbecause if and , so m ustand for , it is the sam e as ., which m akes.If got round up, then all satisfy the condition along withbecause if andCase 1)->Case 2)->and for , since is odd,->->, and is prime so or , which is not in this set, which m akes.Now the only case without rounding, . It must be true.。

2002 AMC 12B Problems and Solution

2002 AMC 12B Problems and Solution Problem 1The arithm etic m ean of the nine numbers in the setis a -digit number , all of whose digits aredistinct. The num ber does not contain the digitSolutionWe wish to find , or. This does not have the digit 0,so the answer isProblem 2What is the value ofwhen ?SolutionBy the distributive property,Problem 3For how m any positive integers is a prime number?SolutionFactoring, we get . Exactly of andmust be and the other a prim e number. If , then , and, which is not prime. On the other hand, if , then ,and , which is a prime number. The answer is .Problem 4Let be a positive integer such that is an integer. Which of the following statem ents is not true:SolutionSince ,From which it follows that and . Thus the answer is.Problem 5Let and be the degree m easures of the five angles of a pentagon.Suppose that and and form an arithmeticsequence. Find the value of .SolutionThe sum of the degree measures of the angles of a pentagon (as a pentagon can besplit into triangles) is . If we let, it follows thatNote that since is the middle term of an arithmetic sequence wit h an odd num ber of term s, it is simply the average of the sequence.Problem 6Suppose that and are nonzero real numbers, and that the equationhas solutions and . Then the pair isSolutionSolution 1Since , it follows bycom paring coefficients that and that . Since is nonzero, ,and . Thus .Solution 2Another method is to use Vieta's formulas. The sum of the solutions to this polynomial is equal to the opposite of the coefficient, since the leading coefficientis 1; in other words, and the product of the solutions is equal to theconstant term (i.e, ). Since is nonzero, it follows that andtherefore (from the first equation), . Hence,The product of three consecutive positive integers is tim es their sum. What is thesum of their squares?SolutionLet the three consecutive positive integers be , , and . So,. Rearranging and factoring,, so . Hence, the sum of the squares is.Problem 8Suppose July of year has five Mondays. Which of the following must occur fivetim es in August of year ? (Note: Both months have 31 days.)SolutionIf there are five Mondays, there are only three possibilities for their dates:, , and .In the first case August starts on a Thursday, and there are five Thursdays, Fridays, and Saturdays in August.In the second case August starts on a Wednesday, and there are five Wednesdays, Thursdays, and Fridays in August.In the third case August starts on a Tuesday, and there are five Tuesdays, Wednesdays, and Thursdays in August.The only day of the week that is guaranteed to appear five tim es is therefore.If are positive real numbers such that form an increasingarithmetic sequence and form a geometric sequence, then isSolutionSolution 1We can let a=1, b=2, c=3, and d=4.Solution 2As is a geom etric sequence, let and for som e .Now, is an arithmetic sequence. Its difference is . Thus.Comparing the two expressions for we get . The positive solution is, and .Solution 3Letting be the common difference of the arithm etic progression, we have, , . We are given that = , orCross-m ultiplying, we getSo .How m any different integers can be expressed as the sum of three distinct m embersof the set ?SolutionEach number in the set is congruent to 1 m odulo 3. Therefore, the sum of any three numbers is a multiple of 3. We can m ake all multiples of three between 1+4+7=12 (the minimu m sum) and 13+16+19=48 (the m aximum sum), inclusive. There areintegers we can form.Problem 11The positive integers and are all prime numbers. The sum ofthese four prim es isSolutionSince and must have the sam e parity, and since there is only oneeven prime number, it follows that and are both odd. Thus one ofis odd and the other even. Since , it follows that(as a prime greater than ) is odd. Thus , and areconsecutive odd primes. At least one of is divisible by , fromwhich it follows that and . The sum of these num bers is thus , aprime, so the answer is .Problem 12For how m any integers is the square of an integer?SolutionSolution 1Let , with (note that the solutions do not give anyadditional solutions for ). Then rewriting, . Since, it follows that divides . Listing the factors of , wefind that are the only solutions (respectively yielding).Solution 2For and the fraction is negative, for it is not defined, and forit is between 0 and 1.Thus we only need to examine and .For and we obviously get the squares and respectively.For prime the fraction will not be an integer, as the denominator will not contain the prime in the numerator.This leaves , and a quick substitution shows that out ofthese only and yield a square.Problem 13The sum of consecutive positive integers is a perfect square. The sm allest possible value of this sum isSolutionLet be the consecutive positive integers. Their sum,, is a perfect square. Since is a perfect square, itfollows that is a perfect square. The sm allest possible such perfect square iswhen , and the sum is .Problem 14Four distinct circles are drawn in a plane. What is the m aximum number of points where at least two of the circles intersect?SolutionFor any given pair of circles, they can intersect at m ost tim es. Since there arepairs of circles, the m aximum number of possible intersections is. We can construct such a situation as below, so the answer is .Problem 15How many four-digit numbers have the property that the three-digit numberobtained by removing the leftm ost digit is one ninth of ?SolutionLet , such that . Then. Since , fromwe have three-digit solutions, and the answer is . Problem 16Juan rolls a fair regular octahedral die marked with the numbers through . ThenAm al rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?SolutionSolution 1On both dice, only the faces with the numbers are divisible by . Letbe the probability that Juan rolls a or a , and thatAm al does. By the Principle of Inclusion-Exclusion,Alternatively, the probability that Juan rolls a m ultiple of is , and the probabilitythat Juan does not roll a multiple of but Amal does is . Thus thetotal probability is .Solution 2The probability that neither Juan nor Amal rolls a m ultiple of is ; usingcom plementary counting, the probability that at least one does is.Problem 17Andy’s lawn has twice as m uch area as Beth’s lawn and three tim es as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and one third as fast as Andy’s mower. If they all start to m ow their lawns at the sam e tim e, who will finish first?SolutionWe say Andy's lawn has an area of . Beth's lawn thus has an area of , andCarlos's lawn has an area of .We say Andy's lawn mower cuts at a speed of . Carlos's cuts at a speed of , andBeth's cuts at a speed .Each person's lawn is cut at a speed of , so Andy's is cut in tim e, as is Carlos's.Beth's is cut in , so the first one to finish is .Problem 18A point is randomly selected from the rectangular region with vertices. What is the probability that is closer to the origin thanit is to the point ?SolutionThe region containing the points closer to than to is bounded by theperpendicular bisector of the segm ent with endpoints . Theperpendicular bisector passes through midpoint of , which is ,the center of the unit square with coordinates . Thus, itcuts the unit square into two equal halves of area . The total area of therectangle is , so the area closer to the origin than to and in the rectangle is. The probability is .Problem 19If and are positive real numbers such thatand , then isSolutionAdding up the three equations gives. Subtracting each of the above equations from this yields, respectively,. Taking their product,. Problem 20Let be a right-angled triangle with . Let and be themidpoints of legs and , respectively. Given that and ,find .SolutionLet , . By the Pythagorean Theorem onrespectively,Summing these gives .By the Pythagorean Theorem again, we haveProblem 21For all positive integers less than , letCalculate .SolutionSince , it follows thatThus .Problem 22For all integers greater than , define . Letand . Then equalsSolutionBy the change of base form ula, . ThusIn , we have and . Side and the m edian from tohave the sam e length. What is ?SolutionLet be the foot of the m edian from to , and we let . Thenby the Law of Cosines on , we haveSince , we can add these two equations and getHence and .Alternate SolutionFrom Stewart's Theorem, we haveSimplifying, we getA convex quadrilateral with area contains a point in its interior suchthat . Find the perimeter of .SolutionWe have (This is true for any convex quadrilateral: split the quadrilateral along and then using the triangle areaformula to evaluate and ), with equality only if . By the triangle inequality,with equality if lies on and respectively. ThusSince we have the equality case, at point , as shown below.By the Pythagorean Theorem,The perimeter of is .Problem 25Let , and let denote the set of points in the coordinateplane such that The area of isclosest toSolutionThe first condition gives us thatwhich is a circle centered at with radius. The second condition gives us thatThus eitherorEach of those lines passes through and has slope , as shown above.Therefore, the area of is half of the area of the circle, which is.。

美国数学竞赛AMC8 -- 2008年真题解析(英文解析+中文解析)

美国数学竞赛AMC8 – 2008年真题解析(英文解析+中文解析)Problem 1Answer: BSolution:50-12-24=14中文解析：总共花的钱是：12+12*2=36元。

剩余50-36=14元。

答案是BProblem 2Answer: ASolution:We can derive that c=8,L=6, U=7,and E=1. Therefore, the answer is 8671.中文解析：这10个字母的对应关系是： B -0；E-1; S-2; ......K -9. 按照这个对应关系：C-8，L-6，U-7，E-1. 即8671. 答案是A。

Problem 3Answer: ASolution:We can go backwards by days, but we can also backwards by weeks. If we go backwards by weeks, we see that February 6 is a Friday. If we now go backwards by days, February 1 is a Sunday.中文解析：13日是周五，则13-7=6，即6日也是周五，则倒推2月1日是周日。

答案是A。

Problem 4Answer: CSolution:The area outside the small triangle but inside the large triangle is 16-1=15. This is equally distributed between the three trapezoids. Each trapezoid has an area of 15/3=5.中文解析：大三角形的面积等于小的等边三角形的面积加上3个梯形的面积。

据此，三个梯形的面积是16-1=15. 每个梯形的面积是15/3=5. 答案是C。

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2008 AMC 12B ProblemsProblem 1A basketball player made baskets during a game. Each basket was worth either or points. How many different numbers could represent the total points scored by the player?SolutionIf the basketball player makes three-point shots and two-point shots, he scores points. Clearly every value of yields a different number of total points. Since he can make any number of three-point shots between and inclusive, the number of different point totals is .Problem 2A block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?SolutionAfter reversing the numbers on the second and fourth rows, the block will look like this:The difference between the two diagonal sums is:.Problem 3A semipro baseball league has teams with players each. League rules state that a player must be paid at least dollars, and that the total of all players' salaries for each team cannot exceed dollars. What is the maximum possiblle salary, in dollars, for a single player?SolutionWe want to find the maximum any player could make, so assume that everyone else makes the minimum possible and that the combined salaries total the maximum ofThe maximum any player could make is dollars (answer choice C)Problem 4On circle , points and are on the same side of diameter , , and . What is the ratio of the area of the smaller sector to the area of the circle?Solution.Since a circle has , the desired ratio is .Problem 5A class collects dollars to buy flowers for a classmate who is in the hospital. Roses cost dollars each, and carnations cost dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly dollars?SolutionThe class could send just carnations (25 of them). They could also send 22 carnations and 2 roses, 19 carnations and 4 roses, and so on, down to 1 carnation and 16 roses. There are 9 total possibilities (from 0 to 16 roses, incrementing by 2 at each step), which is answer choice C.Problem 6Postman Pete has a pedometer to count his steps. The pedometer records up to steps, then flips over to on the next step. Pete plans to determine his mileage for a year. On January Pete sets the pedometer to . During the year, the pedometer flips from to forty-four times. On December the pedometer reads . Pete takes steps per mile. Which of the following is closest to the number of miles Pete walked during the year?SolutionEvery time the pedometer flips, Pete has walked steps. Therefore, Pete has walked a total of steps, which ismiles, which is closest to answer choice A.Problem 7For real numbers and , define . What is ?SolutionProblem 8Points and lie on . The length of is times the length of , and the length of is times the length of . The length of is what fraction of the length of ?SolutionSince and , .Since and , .Thus, .Problem 9Points and are on a circle of radius and . Point is the midpoint of the minor arc . What is the length of the line segment ?SolutionTrig Solution:Let be the angle that subtends the arc AB. By the law of cosines,The half-angle formula says that, which is answer choice A.Other SolutionDefine D as the midpoint of AB, and R the center of the circle. R, C, and D are collinear, and since D is the midpoint of AB, , and so. Since , , and soProblem 10Bricklayer Brenda would take hours to build a chimney alone, and bricklayer Brandon would take hours to build it alone. When they work together they talk a lot, and their combined output is decreased by bricks per hour. Working together, they build the chimney in hours. How many bricks are in the chimney?SolutionLet be the number of bricks in the house.Without talking, Brenda and Brandon lay and bricks per hour respectively, so together they lay per hour together.Since they finish the chimney in hours, . Thus,.Problem 11A cone-shaped mountain has its base on the ocean floor and has a height of 8000feet. The top of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?SolutionIn a cone, radius and height each vary inversely with increasing height (i.e. the radius of the cone formed by cutting off the mountain at feet is half that of the original mountain). Therefore, volume varies as the inverse cube of increasingheight (expressed as a percentage of the total height of cone):Plugging in our given condition,, answer choice A.Problem 12For each positive integer , the mean of the first terms of a sequence is . What is the th term of the sequence?SolutionLetting be the nth partial sum of the sequence:The only possible sequence with this result is the sequence of odd integers.Problem 13Vertex of equilateral is in the interior of unit square . Let be the region consisting of all points inside and outside whosedistance from is between and . What is the area of ?Problem 14A circle has a radius of and a circumference of . What is ?SolutionLet be the circumference of the circle, and let be the radius of the circle. Using log properties, and.Since , .Problem 15 (no solution)On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let be the region formed by the union of the square and all the triangles, and be the smallest convex polygon that contains . What is the area of the region that is inside but outside ?Baidu查的答案答案是A，其实画个图就清楚了，边长为1的正方形（unit square) 连同周边12个正三角形组成一个新的边长为2的正方形，要使包在正方形外面的八边形面积最小，只有A是正确的，BCEDE的话面积都比1/4的时候大。