美国高中数学竞赛 AMC10 2000年试题

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AMC10美国数学竞赛讲义全

AMC 中的数论问题1：Remember the prime between 1 to 100：2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 91 2：Perfect number:Let is the prime number.if 21p - is also the prime number. then 1(21)2pp --is the perfectnumber. For example:6，28，496。

3: Let ,0n abc a =≠ is three digital integer 。

if 333n a b c =++Then the number n is called Daffodils number. There are only four numbers ： 153 370 371 407Let ,0n abcd a =≠ is four digital integer 。

if 4444d n a b c +=++ Then the number n is called Roses number. There are only three numbers: 1634 8208 94744:The Fundamental Theorem of ArithmeticEvery natural number can be written as a product of primes uniquely up to order.5：Suppose that a and b are integers with b =0. Then there exists unique integers q and r such that 0 ≤ r< ｜b ｜ and a = bq + r.6:（1)Greatest Common Divisor: Let gcd (a, b) = max {d ∈ Z： d | a and d | b}。

AMC_美国数学竞赛_2000_AMC_10__试题及答案解析

USA AMC 10 20001In the year , the United States will host the International Mathematical Olympiad. Let , , and be distinct positive integers such that the product . What's the largest possible value of the sum ?SolutionThe sum is the highest if two factors are the lowest.So, and .2Solution.3Each day, Jenny ate of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, remained. How many jellybeans were in the jar originally?Solution4Chandra pays an online service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was , but in January her bill was because she used twice as much connect time as in December. What is the fixxed monthly fee?SolutionLet be the fixed fee, and be the amount she pays for the minutes she used in the first month.We want the fixed fee, which is5Points and are the midpoints of sides and of . As moves along a line that is parallel to side , how many of the four quantities listed below change?(a) the length of the segment(b) the perimeter of(c) the area of(d) the area of trapezoidSolution(a) Clearly does not change, and , so doesn't change either.(b) Obviously, the perimeter changes.(c) The area clearly doesn't change, as both the base and its corresponding height remain the same.(d) The bases and do not change, and neither does the height, so the trapezoid remains the same.Only quantity changes, so the correct answer is .6The Fibonacci Sequence starts with two 1s and each term afterwards is the sum of its predecessors. Which one of the ten digits is the last to appear in thet units position of a number in the Fibonacci Sequence?SolutionThe pattern of the units digits areIn order of appearance:.is the last.7In rectangle , , is on , and and trisect . What is the perimeter of ?Solution.Since is trisected, .Thus,.Adding, .8At Olympic High School, of the freshmen and of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?There are five times as many sophomores as freshmen.There are twice as many sophomores as freshmen.There are as many freshmen as sophomores.There are twice as many freshmen as sophomores.There are five times as many freshmen as sophomores.SolutionLet be the number of freshman and be the number of sophomores.There are twice as many freshmen as sophomores.9If , where , thenSolution, so ...10The sides of a triangle with positive area have lengths , , and . The sides of a second triangle with positive area have lengths , , and . What is the smallest positive number that is not a possible value of ?SolutionFrom the triangle inequality, and . The smallest positive number not possible is , which is .11Two different prime numbers between and are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?SolutionTwo prime numbers between and are both odd.Thus, we can discard the even choices.Both and are even, so one more than is a multiple of four.is the only possible choice.satisfy this, .12Figures , , , and consist of , , , and nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?SolutionSolution 1We have a recursion:.I.E. we add increasing multiples of each time we go up a figure. So, to go from Figure 0 to 100, we add.Solution 2We can divide up figure to get the sum of the sum of the first odd numbers and the sum of the first odd numbers. If you do not see this, here is the example for :The sum of the first odd numbers is , so for figure , there are unit squares. We plug in to get , which is choice13There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?SolutionIn each column there must be one yellow peg. In particular, in the rightmost column, there is only one peg spot, therefore a yellow peg must go there.In the second column from the right, there are two spaces for pegs. One of them is in the same row as the corner peg, so there is only one remaining choice left for the yellow peg in this column.By similar logic, we can fill in the yellow pegs as shown:After this we can proceed to fill in the whole pegboard, so there is only arrangement of the pegs. The answer is14Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were , , , , and . What was the last score Mrs. Walter entered?SolutionThe sum of the first scores must be even, so we must choose evens or the odds to be the first two scores.Let us look at the numbers in mod .If we choose the two odds, the next number must be a multiple of , of which there is none.Similarly, if we choose or , the next number must be a multiple of , of which there is none.So we choose first.The next number must be 1 in mod 3, of which only remains.The sum of the first three scores is . This is equivalent to in mod .Thus, we need to choose one number that is in mod . is the only one that works.Thus, is the last score entered.15Two non-zero real numbers, and , satisfy . Which of the following is a possible value of ?SolutionSubstituting , we get16The diagram shows lattice points, each one unit from its nearest neighbors. Segment meets segment at . Find the length of segment .SolutionSolution 1Let be the line containing and and let be the line containing and . If we set the bottom left point at , then , , , and .The line is given by the equation . The -intercept is , so . We are given two points on , hence we cancompute the slope, to be , so is the lineSimilarly, is given by . The slope in this case is , so . Plugging in the point gives us , so is the line .At , the intersection point, both of the equations must be true, soWe have the coordinates of and , so we can use the distance formula here:which is answer choiceSolution 2Draw the perpendiculars from and to , respectively. As it turns out, . Let be the point on for which ., and , so by AA similarity,By the Pythagorean Theorem, we have ,, and . Let , so , thenThis is answer choiceAlso, you could extend CD to the end of the box and create two similar triangles. Then use ratios and find that the distance is 5/9 of the diagonal AB. Thus, the answer is B.17Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?SolutionConsider what happens each time he puts a coin in. If he puts in a quarter, he gets five nickels back, so the amount of money he has doesn't change. Similarly, if he puts a nickel in the machine, he gets five pennies back and the money value doesn't change. However, if he puts a penny in, he gets five quarters back, increasing the amount of money he has by cents.This implies that the only possible values, in cents, he can have are the ones one more than a multiple of . Of the choices given, the only one is18Charlyn walks completely around the boundary of a square whose sides are each km long. From any point on her path she can see exactly km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?SolutionThe area she sees looks at follows:The part inside the walk has area . The part outside the walk consists of four rectangles, and four arcs. Each of the rectangles has area . The four arcs together form a circle with radius . Therefore the total area she can see is, which rounded to the nearest integer is .19Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the trangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is times the area of the square. The ratio of the area of the other small right triangle to the area of the square isSolutionLet the square have area , then it follows that the altitude of one of the triangles is . The area of the other triangle is .By similar triangles, we haveThis is choice(Note that this approach is enough to get the correct answer in the contest. However, if we wanted a completely correct solution, we should also note that scaling the given triangle times changes each of the areas times, and therefore it does not influence the ratio of any two areas. This is why we can pick the side of the square.)20Let , , and be nonnegative integers such that . What is the maximum value of ?SolutionThe trick is to realize that the sum is similar to the product .If we multiply , we get.We know that , therefore.Therefore the maximum value of is equal to the maximum value of . Now we will find this maximum.Suppose that some two of , , and differ by at least . Then this triple is surely not optimal.Proof: WLOG let . We can then increase the value ofby changing and .Therefore the maximum is achieved in the cases where is a rotation of . The value of in this case is . And thus the maximum of is.21If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?I. All alligators are creepy crawlers.II. Some ferocious creatures are creepy crawlers.III. Some alligators are not creepy crawlers.SolutionWe interpret the problem statement as a query about three abstract concepts denoted as "alligators", "creepy crawlers" and "ferocious creatures". In answering the question, we may NOT refer to reality -- for example to the fact that alligators do exist.To make more clear that we are not using anything outside the problem statement, let's rename the three concepts as , , and .We got the following information:▪If is an , then is an .▪There is some that is a and at the same time an .We CAN NOT conclude that the first statement is true. For example, the situation "Johnny and Freddy are s, but only Johnny is a "meets both conditions, but the first statement is false.We CAN conclude that the second statement is true. We know that there is some that is a and at the same time an . Pick one such and call it Bobby. Additionally, we know that if is an , then is an. Bobby is an , therefore Bobby is an . And this is enough to prove the second statement -- Bobby is an that is also a .We CAN NOT conclude that the third statement is true. For example, consider the situation when , and are equivalent (represent the same set of objects). In such case both conditions are satisfied, but the third statement is false.Therefore the answer is .22One morning each member of Angela's family drank an -ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?SolutionThe exact value "8 ounces" is not important. We will only use the fact that each member of the family drank the same amount.Let be the total number of ounces of milk drank by the family and the total number of ounces of coffee. Thus the whole family drank a total of ounces of fluids.Let be the number of family members. Then each family member drank ounces of fluids.We know that Angela drank ounces of fluids.As Angela is a family member, we have .Multiply both sides by to get .If , we have .If , we have .Therefore the only remaining option is .23When the mean, median, and mode of the list are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ?SolutionAs occurs three times and each of the three other values just once, regardless of what we choose the mode will always be .The sum of all numbers is , therefore the mean is .The six known values, in sorted order, are . From this sequence we conclude: If , the median will be . If , the median will be . Finally, if , the median will be .We will now examine each of these three cases separately.In the case , both the median and the mode are 2, therefore we can not get any non-constant arithmetic progression.In the case we have , because. Therefore our three values inorder are . We want this to be an arithmetic progression. From the first two terms the difference must be . Therefore thethird term must be .Solving we get the only solution for this case: . The case remains. Once again, we have ,therefore the order is . The only solution is when , i. e., .The sum of all solutions is therefore .24Let be a function for which . Find the sum of all values of for which .SolutionIn the definition of , let . We get: . Aswe have , we must have , in other words .One can now either explicitly compute the roots, or use Vieta's formulas. According to them, the sum of the roots of is . In our case this is .(Note that for the above approach to be completely correct, we should additionally verify that there actually are two distinct real roots. This is, for example, obvious from the facts that and .)25In year , the day of the year is a Tuesday. In year , the day is also a Tuesday. On what day of the week did the of year occur?SolutionClearly, identifying what of these years may/must/may not be a leap year will be key in solving the problem.Let be the day of year , the day of year and the day of year .If year is not a leap year, the day will bedays after . As , that would be a Monday.Therefore year must be a leap year. (Then is days after .) As there can not be two leap years after each other, is not a leap year. Therefore day is days after . We have . Therefore is weekdays before , i.e., is a.(Note that the situation described by the problem statement indeed occurs in our calendar. For example, for we have =Tuesday, October 26th 2004, =Tuesday, July 19th, 2005 and =Thursday, April 10th 2003.)。

AMC 美国数学竞赛 2000 AMC 10 试题及答案解析

USA AMC 10 20001In the year , the United States will host the InternationalMathematical Olympiad. Let , , and be distinct positive integerssuch that the product . What's the largest possiblevalue of the sum ?SolutionThe sum is the highest if two factors are the lowest.So, and .2Solution.3Each day, Jenny ate of the jellybeans that were in her jar at thebeginning of the day. At the end of the second day, remained. Howmany jellybeans were in the jar originally?Solution4Chandra pays an online service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was , but inJanuary her bill was because she used twice as much connecttime as in December. What is the fixxed monthly fee?SolutionLet be the fixed fee, and be the amount she pays for the minutes she used in the first month.We want the fixed fee, which is5Points and are the midpoints of sides and of . Asmoves along a line that is parallel to side , how many of the fourquantities listed below change?(a) the length of the segment(b) the perimeter of(c) the area of(d) the area of trapezoidSolution(a) Clearly does not change, and , so doesn'tchange either.(b) Obviously, the perimeter changes.(c) The area clearly doesn't change, as both the base and itscorresponding height remain the same.(d) The bases and do not change, and neither does the height,so the trapezoid remains the same.Only quantity changes, so the correct answer is .6The Fibonacci Sequence starts with two 1s andeach term afterwards is the sum of its predecessors. Which one of the ten digits is the last to appear in thet units position of a number in the Fibonacci Sequence?SolutionThe pattern of the units digits areIn order of appearance:.is the last.7In rectangle , , is on , and and trisect. What is the perimeter of ?Solution.Since is trisected, .Thus,.Adding, .8At Olympic High School, of the freshmen and of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?There are five times as many sophomores as freshmen.There are twice as many sophomores as freshmen.There are as many freshmen as sophomores.There are twice as many freshmen as sophomores.There are five times as many freshmen as sophomores.Let be the number of freshman and be the number of sophomores.There are twice as many freshmen as sophomores.9If , where , thenSolution, so ...10The sides of a triangle with positive area have lengths , , and .The sides of a second triangle with positive area have lengths , ,and . What is the smallest positive number that is not a possiblevalue of ?SolutionFrom the triangle inequality, and . The smallestpositive number not possible is , which is .11Two different prime numbers between and are chosen. When theirsum is subtracted from their product, which of the following numbers could be obtained?Two prime numbers between and are both odd.Thus, we can discard the even choices.Both and are even, so one more than is a multipleof four.is the only possible choice.satisfy this, .12Figures , , , and consist of , , , and nonoverlapping unitsquares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?SolutionSolution 1We have a recursion:.I.E. we add increasing multiples of each time we go up a figure.So, to go from Figure 0 to 100, we add.Solution 2We can divide up figure to get the sum of the sum of the firstodd numbers and the sum of the first odd numbers. If you do not see this, here is the example for :The sum of the first odd numbers is , so for figure , there areunit squares. We plug in to get , which ischoice13There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?SolutionIn each column there must be one yellow peg. In particular, in the rightmost column, there is only one peg spot, therefore a yellow peg must go there.In the second column from the right, there are two spaces for pegs. One of them is in the same row as the corner peg, so there is only one remaining choice left for the yellow peg in this column.By similar logic, we can fill in the yellow pegs as shown:After this we can proceed to fill in the whole pegboard, so there is onlyarrangement of the pegs. The answer is14Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were , ,, , and . What was the last score Mrs. Walter entered?The sum of the first scores must be even, so we must choose evensor the odds to be the first two scores.Let us look at the numbers in mod .If we choose the two odds, the next number must be a multiple of ,of which there is none.Similarly, if we choose or , the next number must be amultiple of , of which there is none.So we choose first.The next number must be 1 in mod 3, of which only remains.The sum of the first three scores is . This is equivalent to in mod.Thus, we need to choose one number that is in mod . is the onlyone that works.Thus, is the last score entered.15Two non-zero real numbers, and , satisfy . Which of thefollowing is a possible value of ?SolutionSubstituting , we get16The diagram shows lattice points, each one unit from its nearestneighbors. Segment meets segment at . Find the length ofsegment .SolutionSolution 1Let be the line containing and and let be the line containingand . If we set the bottom left point at , then ,, , and .The line is given by the equation . The -intercept is, so . We are given two points on , hence we cancompute the slope, to be , so is the lineSimilarly, is given by . The slope in this case is ,so . Plugging in the point gives us , so is theline .At , the intersection point, both of the equations must be true, soWe have the coordinates of and , so we can use the distanceformula here:which is answer choiceSolution 2Draw the perpendiculars from and to , respectively. As it turnsout, . Let be the point on for which ., and , so by AA similarity,By the Pythagorean Theorem, we have ,, and . Let , so, thenThis is answer choiceAlso, you could extend CD to the end of the box and create two similar triangles. Then use ratios and find that the distance is 5/9 of the diagonal AB. Thus, the answer is B.17Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?SolutionConsider what happens each time he puts a coin in. If he puts in a quarter, he gets five nickels back, so the amount of money he has doesn't change. Similarly, if he puts a nickel in the machine, he gets five pennies back and the money value doesn't change. However, if he puts a penny in, he gets five quarters back, increasing the amount of money he has by cents.This implies that the only possible values, in cents, he can have are the ones one more than a multiple of . Of the choices given, theonly one is18Charlyn walks completely around the boundary of a square whose sides are each km long. From any point on her path she can seeexactly km horizontally in all directions. What is the area of theregion consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?SolutionThe area she sees looks at follows:The part inside the walk has area . The part outside thewalk consists of four rectangles, and four arcs. Each of the rectangles has area . The four arcs together form a circle with radius .Therefore the total area she can see is, which rounded to the nearest integeris .19Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the trangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is times the area of the square. The ratio of the area of the other small right triangle to the area of the square isSolutionLet the square have area , then it follows that the altitude of one ofthe triangles is . The area of the other triangle is .By similar triangles, we haveThis is choice(Note that this approach is enough to get the correct answer in the contest. However, if we wanted a completely correct solution, we should also note that scaling the given triangle times changes eachof the areas times, and therefore it does not influence the ratio ofany two areas. This is why we can pick the side of the square.)20Let , , and be nonnegative integers such that .What is the maximum value of ?SolutionThe trick is to realize that the sum is similarto the product .If we multiply , we get.We know that , therefore.Therefore the maximum value of is equal tothe maximum value of . Now we will find this maximum.Suppose that some two of , , and differ by at least . Then thistriple is surely not optimal.Proof: WLOG let . We can then increase the value ofby changing and .Therefore the maximum is achieved in the cases where is arotation of . The value of in this case is . And thus the maximum of is.21If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?I. All alligators are creepy crawlers.II. Some ferocious creatures are creepy crawlers.III. Some alligators are not creepy crawlers.SolutionWe interpret the problem statement as a query about three abstract concepts denoted as "alligators", "creepy crawlers" and "ferocious creatures". In answering the question, we may NOT refer to reality -- for example to the fact that alligators do exist.To make more clear that we are not using anything outside the problem statement, let's rename the three concepts as , , and .We got the following information:▪If is an , then is an .▪There is some that is a and at the same time an .We CAN NOT conclude that the first statement is true. For example, the situation "Johnny and Freddy are s, but only Johnny is a "meets both conditions, but the first statement is false.We CAN conclude that the second statement is true. We know that there is some that is a and at the same time an . Pick one suchand call it Bobby. Additionally, we know that if is an , then is an. Bobby is an , therefore Bobby is an . And this is enough toprove the second statement -- Bobby is an that is also a .We CAN NOT conclude that the third statement is true. For example, consider the situation when , and are equivalent (represent thesame set of objects). In such case both conditions are satisfied, but the third statement is false.Therefore the answer is .22One morning each member of Angela's family drank an -ouncemixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?SolutionThe exact value "8 ounces" is not important. We will only use the fact that each member of the family drank the same amount.Let be the total number of ounces of milk drank by the family and the total number of ounces of coffee. Thus the whole family drank a total of ounces of fluids.Let be the number of family members. Then each family memberdrank ounces of fluids.We know that Angela drank ounces of fluids.As Angela is a family member, we have .Multiply both sides by to get .If , we have .If , we have .Therefore the only remaining option is .23When the mean, median, and mode of the list arearranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ?SolutionAs occurs three times and each of the three other values just once,regardless of what we choose the mode will always be .The sum of all numbers is , therefore the mean is .The six known values, in sorted order, are . From this sequence we conclude: If , the median will be . If , themedian will be . Finally, if , the median will be .We will now examine each of these three cases separately.In the case , both the median and the mode are 2, therefore wecan not get any non-constant arithmetic progression.In the case we have , because. Therefore our three values inorder are . We want this to be an arithmetic progression. From the first two terms the difference must be . Therefore thethird term must be .Solving we get the only solution for this case: .The case remains. Once again, we have ,therefore the order is . The only solution is when , i. e., .The sum of all solutions is therefore .24Let be a function for which . Find the sum of allvalues of for which .SolutionIn the definition of , let . We get: . Aswe have , we must have , in other words.One can now either explicitly compute the roots, or use Vieta'sformulas. According to them, the sum of the roots ofis . In our case this is .(Note that for the above approach to be completely correct, we should additionally verify that there actually are two distinct real roots. Thisis, for example, obvious from the facts that and .)25In year , the day of the year is a Tuesday. In year , theday is also a Tuesday. On what day of the week did the ofyear occur?SolutionClearly, identifying what of these years may/must/may not be a leap year will be key in solving the problem.Let be the day of year , the day of year andthe day of year .If year is not a leap year, the day will bedays after . As , that would be a Monday.Therefore year must be a leap year. (Then is days after .)As there can not be two leap years after each other, is not a leapyear. Therefore day is days after . We have. Therefore is weekdays before , i.e., is a.(Note that the situation described by the problem statement indeed occurs in our calendar. For example, for we have=Tuesday, October 26th 2004, =Tuesday, July 19th, 2005 and=Thursday, April 10th 2003.)。

2000-2013 mcm美国大学生数学建模竞赛原版题目

2000 Mathematical Contest in ModelingThe ProblemsProblem A: Air traffic ControlProblem B: Radio Channel AssignmentsProblem A Air traffic ControlDedicated to the memory of Dr. Robert Machol, former chief scientist of the Federal Aviation AgencyTo improve safety and reduce air traffic controller workload, the Federal Aviation Agency (FAA) is considering adding software to the air traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA has posed the following problems.Requirement A: Given two airplanes flying in space, when should the air traffic controller consider the objects to be too close and to require intervention?Requirement B: An airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how do we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of aircraft simultaneously passing through that sector (1) at any one instant? (2) during any given interval of time?(3) during a particular time of day? How does the number of potential conflicts arising during those periods affect complexity?Does the presence of additional software tools to automatically predict conflicts and alert the controller reduce or add to this complexity?In addition to the guidelines for your report, write a summary (no more than two pages) that the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your conclusions.Problem BRadio Channel AssignmentsWe seek to model the assignment of radio channels to a symmetric network of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grid (honeycomb-style), as shown in Figure 1, where a transmitter is located at the center of each hexagon.Figure 1An interval of the frequency spectrum is to be allotted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1, 2, 3, ... . Each transmitter will be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided. Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assign channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span.Let s be the length of a side of one of the hexagons. We concentrate on the case that there are two levels of interference.Requirement A: There are several constraints on frequency assignments. First, no two transmitters within distance 4s of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these constraints, what can we say about the span in,Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in all directions.Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance 2s differ by at least some given integer k, while those at distance at most 4s must still differ by at least one. What can we say about the span and about efficient strategies for designing assignments, as a function of k?Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider?Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining your findings.2001 Mathematical Contest in ModelingThe ProblemsProblem A: Choosing a Bicycle WheelProblem B: Escaping a Hurricane's Wrath (An Ill Wind...)Problem A: Choosing a Bicycle WheelCyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike.Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course.Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on theright.The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear.∙Task 1. Provide a table giving the wind speed at which the power required for a solid rear wheel is less than for a spoked rear wheel. The table should include the windspeeds for different road grades starting from zero percent to ten percent in onepercent increments. (Road grade is defined to be the ratio of the total rise of a hilldivided by the length of the road. If the hill is viewed as a triangle, the grade is the sine of the angle at the bottom of the hill.) A rider starts at the bottom of the hill at a speed of 45 kph, and the deceleration of the rider is proportional to the road grade. A riderwill lose about 8 kph for a five percent grade over 100 meters.∙Task 2. Provide an example of how the table could be used for a specific time trial course.∙Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision.Problem B: Escaping a Hurricane's Wrath (An Ill Wind...)Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare.The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading inland from Myrtle Beach and Hilton Head is also planned.A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of the coastal strip. (More accurate data, if sought, are widely available.)The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three. Columbia, another metro area of around 500,000 people, does not have sufficient hotel space to accommodate the evacuees (including some coming from farther north by other routes), so some traffic continues outbound on I-26 towards Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. In 1999, traffic leaving Columbia going northwest was moving only very slowly. Construct a model for the problem to investigate what strategies may reduce the congestion observed in 1999. Here are the questions that need to be addressed:1.Under what conditions does the plan for turning the two coastal-bound lanes of I-26into two lanes of Columbia-bound traffic, essentially turning the entire I-26 intoone-way traffic, significantly improve evacuation traffic flow?2.In 1999, the simultaneous evacuation of the state's entire coastal region was ordered.Would the evacuation traffic flow improve under an alternative strategy that staggers the evacuation, perhaps county-by-county over some time period consistent with thepattern of how hurricanes affect the coast?3.Several smaller highways besides I-26 extend inland from the coast. Under whatconditions would it improve evacuation flow to turn around traffic on these?4.What effect would it have on evacuation flow to establish more temporary shelters inColumbia, to reduce the traffic leaving Columbia?5.In 1999, many families leaving the coast brought along their boats, campers, andmotor homes. Many drove all of their cars. Under what conditions should there berestrictions on vehicle types or numbers of vehicles brought in order to guaranteetimely evacuation?6.It has been suggested that in 1999 some of the coastal residents of Georgia and Florida,who were fleeing the earlier predicted landfalls of Hurricane Floyd to the south, came up I-95 and compounded the traffic problems. How big an impact can they have on the evacuation traffic flow?Clearly identify what measures of performance are used to comparestrategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.Clearly identify what measures of performance are used to compare strategies.Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.2002 Mathematical Contest in ModelingThe ProblemsProblem AAuthors: Tjalling YpmaTitle: Wind and WatersprayAn ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area.Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.Authors: Bill Fox and Rich WestTitle: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat.Historically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook-that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations.Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.Consider the overbooking issue in light of the current situation:Less flights by airlines from point A to point BHeightened security at and around airportsPassengers' fearLoss of billions of dollars in revenue by airlines to dateBuild a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the company's revenue is maximized. Insure that your model reflects the issues above, and consider alternatives for handling "bumped" passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.2003 MCM ProblemsPROBLEM A: The Stunt PersonAn exciting action scene in a movie is going to be filmed, and you are the stunt coordinator! A stunt person on a motorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. You need to protect the stunt person, and also use relatively few cardboard boxes (lower cost, not seen by camera, etc.).∙determine what size boxes to use∙determine how many boxes to use∙determine how the boxes will be stacked∙determine if any modifications to the boxes would help∙generalize to different combined weights (stunt person & motorcycle) and different jump heightsNote that, in "Tomorrow Never Dies", the James Bond character on a motorcycle jumps over a helicopter.PROBLEM B: Gamma Knife Treatment PlanningStereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. Three modalities are commonly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators.The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy helmet. All 201 beams simultaneously intersect at the isocenter, resulting in a spherical (approximately) dose distribution at the effective dose levels. Irradiating the isocenter to deliver dose is termed a “shot.” Shots can be represented as different spheres. Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. For a target volume larger than one shot, multiple shots can be used to cover the entire target. In practice, most target volumes are treated with 1 to 15 shots. The target volume is a bounded, three-dimensional digital image that usually consists of millions of points.The goal of radiosurgery is to deplete tumor cells while preserving normal structures. Since there are physical limitations and biological uncertainties involved in this therapy process, a treatment plan needs to account for all those limitations and uncertainties. In general, an optimal treatment plan is designed to meet the following requirements.1.Minimize the dose gradient across the target volume.2.Match specified isodose contours to the target volumes.3.Match specified dose-volume constraints of the target and critical organ.4.Minimize the integral dose to the entire volume of normal tissues or organs.5.Constrain dose to specified normal tissue points below tolerance doses.6.Minimize the maximum dose to critical volumes.In gamma unit treatment planning, we have the following constraints:1.Prohibit shots from protruding outside the target.2.Prohibit shots from overlapping (to avoid hot spots).3.Cover the target volume with effective dosage as much as possible. But at least 90% ofthe target volume must be covered by shots.e as few shots as possible.Your tasks are to formulate the optimal treatment planning for a gamma knife unit as a sphere-packing problem, and propose an algorithm to find a solution. While designing your algorithm, you must keep in mind that your algorithm must be reasonably efficient.2004 MCM ProblemsPROBLEM A: Are Fingerprints Unique?It is a commonplace belief that the thumbprint of every human who has ever lived is different. Develop and analyze a model that will allow you to assess the probability that this is true. Compare the odds (that you found in this problem) of misidentification by fingerprint evidence against the odds of misidentification by DNA evidence.PROBLEM B: A Faster QuickPass System"QuickPass" systems are increasingly appearing to reduce people's time waiting in line, whether it is at tollbooths, amusement parks, or elsewhere. Consider the design of a QuickPass system for an amusement park. The amusement park has experimented by offering QuickPasses for several popular rides as a test. The idea is that for certain popular rides you can go to a kiosk near that ride and insert your daily park entrance ticket, and out will come a slip that states that you can return to that ride at a specific time later. For example, you insert your daily park entrance ticket at 1:15 pm, and the QuickPass states that you can come back between 3:30 and 4:30 pm when you can use your slip to enter a second, and presumably much shorter, line that will get you to the ride faster. To prevent people from obtaining QuickPasses for several rides at once, the QuickPass machines allow you to have only one active QuickPass at a time.You have been hired as one of several competing consultants to improve the operation of QuickPass. Customers have been complaining about some anomalies in the test system. For example, customers observed that in one instance QuickPasses were being offered for a return time as long as 4 hours later. A short time later on the same ride, the QuickPasses were given for times only an hour or so later. In some instances, the lines for people with Quickpasses are nearly as long and slow as the regular lines.The problem then is to propose and test schemes for issuing QuickPasses in order to increase people's enjoyment of the amusement park. Part of the problem is to determine what criteria to use in evaluating alternative schemes. Include in your report a non-technical summary for amusement park executives who must choose between alternatives from competing consultants.2005 MCM ProblemsPROBLEM A: Flood PlanningLake Murray in central South Carolina is formed by a large earthen dam, which was completed in 1930 for power production. Model the flooding downstream in the event there is a catastrophic earthquake that breaches the dam.Two particular questions:Rawls Creek is a year-round stream that flows into the Saluda River a short distance downriver from the dam. How much flooding will occur in Rawls Creek from a dam failure, and how far back will it extend?Could the flood be so massive downstream that water would reach up to the S.C. State Capitol Building, which is on a hill overlooking the Congaree River?PROBLEM B: TollboothsHeavily-traveled toll roads such as the Garden State Parkway , Interstate 95, and so forth, are multi-lane divided highways that are interrupted at intervals by toll plazas. Because collecting tolls is usually unpopular, it is desirable to minimize motorist annoyance by limiting the amount of traffic disruption caused by the toll plazas. Commonly, a much larger number of tollbooths is provided than the number of travel lanes entering the toll plaza. Upon entering the toll plaza, the flow of vehicles fans out to the larger number of tollbooths, and when leaving the toll plaza,the flow of vehicles is required to squeeze back down to a number of travel lanes equal to the number of travel lanes before the toll plaza. Consequently, when traffic is heavy, congestion increases upon departure from the toll plaza. When traffic is very heavy, congestion also builds at the entry to the toll plaza because of the time required for each vehicle to pay the toll.Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane. Under what conditions is this more or less effective than the current practice? Note that the definition of "optimal" is up to you to determine.2005 ICM ProblemPROBLEM C: Nonrenewable ResourcesSelect a vital nonrenewable or exhaustible resource (water, mineral, energy, food, etc.) for which your team can find appropriate world-wide historic data on its endowment, discovery, annual consumption, and price.The modeling tasks are:ing the endowment, discoveries, and consumption data, model the depletion ordegradation of the commodity over a long horizon using resource modeling principles.2.Adjust the model to account for future economic, demographic, political andenvironmental factors. Be sure to reveal the details of your model, provide visualizations of the model’s output, and explain limitations of the model.3.Create a fair, practical "harvesting/management" policy that may include economicincentives or disincentives, which sustain the usage over a long period of time while avoiding severe disruption of consumption, degradation or rapid exhaustion of the resource.4.Develop a "security" policy that protects the resource against theft, misuse, disruption,and unnecessary degradation or destruction of the resource. Other issues that may need to be addressed are political and security management alternatives associated with these policies.5.Develop policies to control any short- or long-term "environmental effects" of theharvesting. Be sure to include issues such as pollutants, increased susceptibility to natural disasters, waste handling and storage, and other factors you deem appropriate.pare this resource with any other alternatives for its purpose. What new science ortechnologies could be developed to mitigate the use and potential exhaustion of this resource? Develop a research policy to advance these new areas2006 MCM ProblemsPROBLEM A: Positioning and Moving Sprinkler Systems for IrrigationThere are a wide variety of techniques available for irrigating a field. The technologies range from advanced drip systems to periodic flooding. One of the systems that is used on smaller ranches is the use of "hand move" irrigation systems. Lightweight aluminum pipes with sprinkler heads are put in place across fields, and they are moved by hand at periodic intervals to insure that the whole field receives an adequate amount of water. This type of irrigation system is cheaper and easier to maintain than other systems. It is also flexible, allowing for use on a wide variety of fields and crops. The disadvantage is that it requires a great deal of time and effort to move and set up the equipment at regular intervals.Given that this type of irrigation system is to be used, how can it be configured to minimize the amount of time required to irrigate a field that is 80 meters by 30 meters? For this task you are asked to find an algorithm to determine how to irrigate the rectangular field that minimizes the amount of time required by a rancher to maintain the irrigation system. One pipe set is used in the field. You should determine the number of sprinklers and the spacing between sprinklers, and you should find a schedule to move the pipes, including where to move them.A pipe set consists of a number of pipes that can be connected together in a straight line. Each pipe has a 10 cm inner diameter with rotating spray nozzles that have a 0.6 cm inner diameter. When put together the resulting pipe is 20 meters long. At the water source, the pressure is 420 Kilo- Pascal's and has a flow rate of 150 liters per minute. No part of the field should receive more than 0.75 cm per hour of water, and each part of the field should receive at least 2 centimeters of water every 4 days. The total amount of water should be applied as uniformly as possiblePROBLEM B: Wheel Chair Access at AirportsOne of the frustrations with air travel is the need to fly through multiple airports, and each stop generally requires each traveler to change to a different airplane. This can be especially difficult for people who are not able to easily walk to a different flight's waiting area. One of the ways that an airline can make the transition easier is to provide a wheel chair and an escort to those people who ask for help. It is generally known well in advance which passengers require help, but it is not uncommon to receive notice when a passenger first registers at the airport. In rare instances an airline may not receive notice from a passenger until just prior to landing.Airlines are under constant pressure to keep their costs down. Wheel chairs wear out and are expensive and require maintenance. There is also a cost for making the escorts available. Moreover, wheel chairs and their escorts must be constantly moved around the airport so that they are available to people when their flight lands. In some large airports the time required to move across the airport is nontrivial. The wheel chairs must be stored somewhere, but space is expensive and severely limited in an airport terminal. Also, wheel chairs left in high traffic areas represent a liability risk as people try to move around them. Finally, one of the biggest costs is the cost of holding a plane if someone must wait for an escort and becomes late for their flight. The latter cost is especially troubling because it can affect the airline's average flight delay which can lead to fewer ticket sales as potential customers may choose to avoid an airline.Epsilon Airlines has decided to ask a third party to help them obtain a detailed analysis of the issues and costs of keeping and maintaining wheel chairs and escorts available for passengers. The airline needs to find a way to schedule the movement of wheel chairs throughout each day in a cost effective way. They also need to find and define the costs for budget planning in both the short and long term.Epsilon Airlines has asked your consultant group to put together a bid to help them solve their problem. Your bid should include an overview and analysis of the situation to help them decide if you fully understand their problem. They require a detailed description of an algorithm that you would like to implement which can determine where the escorts and wheel chairs should be and how they should move throughout each day. The goal is to keep the total costs as low as possible. Your bid is one of many that the airline will consider. You must make a strong case as to why your solution is the best and show that it will be able to handle a wide range of airports under a variety of circumstances.Your bid should also include examples of how the algorithm would work for a large (at least 4 concourses), a medium (at least two concourses), and a small airport (one concourse) under high and low traffic loads. You should determine all potential costs and balance their respective weights. Finally, as populations begin to include a higher percentage of older people who have more time to travel but may require more aid, your report should include projections of potential costs and needs in the future with recommendations to meet future needs.2007 MCM ProblemsPROBLEM A: GerrymanderingThe United States Constitution provides that the House of Representatives shall be composed of some number (currently 435) of individuals who are elected from each state in proportion to the state's population relative to that of the country as a whole. While this provides a way of determining how many representatives each state will have, it says nothing about how the district represented by a particular representative shall be determined geographically. This oversight has led to egregious (at least some people think so, usually not the incumbent) district shapes that look "unnatural" by some standards.Hence the following question: Suppose you were given the opportunity to draw congressional districts for a state. How would you do so as a purely "baseline" exercise to create the "simplest" shapes for all the districts in a state? The rules include only that each district in the state must contain the same population. The definition of "simple" is up to you; but you need to make a convincing argument to voters in the state that your solution is fair. As an application of your method, draw geographically simple congressional districts for the state of New York.PROBLEM B: The Airplane Seating ProblemAirlines are free to seat passengers waiting to board an aircraft in any order whatsoever. It has become customary to seat passengers with special needs first, followed by first-class passengers (who sit at the front of the plane). Then coach and business-class passengers are seated by groups of rows, beginning with the row at the back of the plane and proceeding forward.Apart from consideration of the passengers' wait time, from the airline's point of view, time is money, and boarding time is best minimized. The plane makes money for the airline only when it is in motion, and long boarding times limit the number of trips that a plane can make in a day.The development of larger planes, such as the Airbus A380 (800 passengers), accentuate the problem of minimizing boarding (and deboarding) time.Devise and compare procedures for boarding and deboarding planes with varying numbers of passengers: small (85-210), midsize (210-330), and large (450-800).。

美国AMC10中文版试题及答案

2000到20XX年AMC10美国数学竞赛0 0P 0 A 0 B 0 C 0D 0 全美中学数学分级能力测验(AMC 10)2000年第01届美国AMC10 (2000年2月日时间75分钟)1. 国际数学奥林匹亚将于在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ⨯M ⨯O =2001，则试问I +M +O 之最大值为。

(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为。

(A) 40 (B) 50 (C) 55 (D) 60 (E) 754. Candra 每月要付给网络公司固定的月租费及上网的拨接费，已知她12月的账单为12.48元，而她1月的账单为17.54元，若她1月的上网时间是12月的两倍，试问月租费是元。

(A) 2.53 (B) 5.06 (C) 6.24 (D) 7.42 (E) 8.775. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直在线移动时，下列各数值有项会变动。

(a) MN 长 (b) △P AB 之周长 (c) △P AB 之面积 (d) ABNM 之面积(A) 0项 (B) 1项 (C) 2项 (D) 3项 (E) 4项 6. 费氏数列是以两个1开始，接下来各项均为前两项之和，试问在费氏数列各项的个位数字中，最后出现的阿拉伯数字为。

(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分∠ADC ，试问△BDP 之周长为。

2000年全国高中数学联合竞赛试卷及参考

2000年全国高中数学联合竞赛试卷及参考答案（10月15日上午8:00-9:40）一、选择题（本题满分36分，每小题6分）1．设全集是实数，若A ={x |2-x ≤0}，B ={x |2210-x =x 10},则B A 是( )(A){2} (B){-1} (C){x |x ≤2} (D)∅2．设sin α＞0,cos α＜0,且sin 3α＞cos 3α,则3α的取值范围是( ) (A)(2k π+6π,2k π+3π), k ∈Z (B)(32πk +6π,32πk +3π),k ∈Z (C)(2k π+65π,2k π+π),k ∈Z (D)(2k π+4π,2k π+3π) (2k π+65π,2k π+π),k ∈Z 3．已知点A 为双曲线x 2-y 2=1的左顶点，点B 和点C 在双曲线的右分支上，△ABC 是等边三角形，则△ABC 的面积是( ) (A)33 (B)233 (C)33 (D)63 4．给定正数p ,q ,a ,b ,c ，其中p ≠q ，若p ,a ,q 是等比数列，p ,b ,c ,q 是等差数列，则一元二次方程bx 2-2ax +c =0( )(A)无实根 (B)有两个相等实根 (C)有两个同号相异实根 (D)有两个异号实根5．平面上整点（纵、横坐标都是整数的点）到直线5435+=x y 的距离中的最小值是( ) (A)17034 (B)8534 (C)201 (D)301 6．设5sin 5cosππωi +=，则以ω,ω3,ω7,ω9为根的方程是( )(A)x 4+x 3+x 2+x +1=0 (B) x 4-x 3+x 2-x +1=0(C) x 4-x 3-x 2+x +1=0 (D) x 4+x 3+x 2-x -1=0 二、填空题（本题满分54分，每小题9分）7．arcsin(sin2000︒)=__________.8．设a n 是(3-nx )的展开式中x 项的系数(n =2,3,4,…)，则nnn a a a 333(lim 3322+++∞→ )=________. 9．等比数列a +log 23,a +log 43,a +log 83的公比是____________.10．在椭圆12222=+b y a x (a ＞b ＞0)中，记左焦点为F ，右顶点为A ，短轴上方的端点为B .若该椭圆的离心率是215-，则∠ABF =_________. 11．一个球与正四面体的六条棱都相切，若正四面体的棱长为a ，则这个球的体积是________.12．如果：(1)a ,b ,c ,d 都属于{1,2,3,4};(2)a ≠b ,b ≠c ,c ≠d ,d ≠a ;(3)a 是a ,b ,c ,d 中的最小值，那么，可以组成的不同的四位数abcd 的个数是_________.三、解答题(本题满分60分，每小题20分)13．设S n =1+2+3+…+n ,n ∈N ，求f (n )=1)32(++n n S n S 的最大值.14．若函数21321)(2+-=x x f 在区间[a ,b ]上的最小值为2a ，最大值为2b ，求[a ,b ].15．已知C 0:x 2+y 2=1和C 1:12222=+by a x (a ＞b ＞0)。

AMC10美国数学竞赛真题 xx年

AMC10美国数学竞赛真题 xx年Problem 1The median of the lististhe mean?Solution. What isProblem 2A number is more than the product of its reciprocal and its additive inverse. In which interval does the number lie?SolutionProblem 3The sum of two numbers is . Suppose is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?SolutionProblem 4What is the maximum number for the possible points of intersection of a circle and a triangle?SolutionProblem 5How many of the twelve pentominoes pictured below have at least one line of symmetry?SolutionProblem 6Letanddenote the product and the sum, respectively, of the digits ofand. Supposeis athe integer . For example,two-digit number such that . What is the units digit ofSolutionProblem 7When the decimal point of a certain positive decimal number is moved four places to the right, thenew number is four times the reciprocal of the original number. What is the original number?SolutionProblem 8Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Theirschedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?SolutionProblem 9The state income tax where Kristin lives is levied at the rate ofof annual income plusof any amount aboveof the first. Kristin of hernoticed that the state income tax she paid amounted to annual income. What was her annual income?Problem 10If ,, and are positive withis,, andSolution, thenProblem 11Consider the dark square in an array of unit squares, part of which is shown. The ?rst ring of squares around this center square contains unit squares. The second ring contains unit squares. If we continue this process, the number of unit squares in the ring isSolutionProblem 12Suppose that is the product of three consecutive integers and that by . Which of the following is not necessarily a divisor of ?is divisibleProblem 13A telephone number has the form , where each letter represents a different digit. The digits in each partof the numbers are in decreasing order; that is, , , and . Furthermore, , , and are consecutive even digits; , , , and are consecutive odd digits; and . Find . SolutionProblem 14A charity sells benefit tickets for a total of . Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?SolutionProblem 15A street has parallel curbs feet apart. A crosswalk bounded by two parallel stripes crosses the street atan angle. The length of the curb between the stripesis feet and each stripe is feet long. Find the distance,in feet, between the stripes?SolutionProblem 16The mean of three numbers is more than the least of the numbers and less than the greatest. The median of the three numbers is . What is their sum?SolutionProblem 17Which of the cones listed below can be formed froma radius by aligning the two straight sides?sector of a circle ofSolutionProblem 18The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest toSolutionProblem 19Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible? SolutionProblem 20A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length . What is the length of each side of the octagon?SolutionProblem 21A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter and altitude , and the axes of the cylinder and cone coincide. Find the radius of the cylinder.SolutionProblem 22In the magic square shown, the sums of the numbersin each row, column, and diagonal are the same. Five of these numbers are represented by , , , , and . Find .SolutionProblem 23A box contains exactly five chips, three red and two white. Chips are randomly removed one at a timewithout replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?SolutionProblem 24In trapezoid, ,andare perpendicular to , and . What is, withSolutionProblem 25How many positive integers not exceeding ?are multiples of or but notSolutionProblem 1The median of the lististhe mean?Solution. What isProblem 2A number is more than the product of its reciprocal and its additive inverse. In which interval does the number lie?SolutionProblem 3The sum of two numbers is . Suppose is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?SolutionProblem 4What is the maximum number for the possible points of intersection of a circle and a triangle?SolutionProblem 5How many of the twelve pentominoes pictured below have at least one line of symmetry?SolutionProblem 6Letanddenote the product and the sum, respectively, of the digits ofand. Supposeis athe integer . For example,two-digit number such that . What is the units digit ofSolutionProblem 7When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?SolutionProblem 8Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Theirschedule is as follows: Darren works every thirdschool day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?SolutionProblem 9The state income tax where Kristin lives is levied at the rate ofof annual income plusof any amount aboveof the first. Kristin of hernoticed that the state income tax she paid amounted to annual income. What was her annual income?SolutionProblem 10If ,, and are positive withis,, and, thenProblem 11Consider the dark square in an array of unit squares, part of which is shown. The ?rst ring of squares around this center square contains unit squares. The second ring contains unit squares. If we continue this process, the number of unit squares in the ring isSolutionProblem 12Suppose that is the product of three consecutive integers and that by . Which of the following is not necessarily a divisor of ?is divisibleSolutionProblem 13A telephone number has the form , where each letter represents a different digit. The digits in each partof the numbers are in decreasing order; that is, , , and . Furthermore, , , and are consecutive even digits; , , , and are consecutive odd digits; and . Find .Problem 14A charity sells benefit tickets for a total of . Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?SolutionProblem 15A street has parallel curbs feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is feet and each stripe is feet long. Find the distance, in feet, between the stripes?SolutionProblem 16The mean of three numbers is more than the least of the numbers and less than the greatest. The median of the three numbers is . What is their sum?SolutionProblem 17Which of the cones listed below can be formed from a radius by aligning the two straight sides?sector of a circle ofSolutionProblem 18The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest toSolutionProblem 19Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible? SolutionProblem 20A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length . What is the length of each side of the octagon?。

AMC10美国数学竞赛讲义全

3: Let ,0n abc a =≠ is three digital integer 。

if 333n a b c =++Then the number n is called Daffodils number. There are only four numbers ： 153 370 371 407Let ,0n abcd a =≠ is four digital integer 。

美国AMC10中文版试题及答案

(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为。

(A) 2.53 (B) 5.06 (C) 6.24 (D) 7.42 (E) 8.775. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直在线移动时，下列各数值有项会变动。

(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分∠ADC ，试问△BDP 之周长为。

AMC10的真题答案及中文翻译

AMC10的真题答案及中文翻译
1、一张展览票原价为20美元。

Susan使用优惠券购买4张票，享受75折优惠。

Pam使用优惠券购买5张票，享受7折优惠。

Pam比Susan多花了多少美元？
2、一个养鱼缸的底部为矩形，长100cm，宽40cm，高50cm。

水位高度为40cm。

将一个底部为矩形，长40cm，宽20cm，高10cm的砖块放入缸中。

水位上升了多少厘米？
3、两个连续的奇数中，较大的数是较小的数的3倍。

它们的和是多少？
4、一个学校商店售卖7支铅笔和8个笔记本需要4.15美元，售卖5支铅笔和3个笔记本需要1.77美元。

那么16支铅笔和10个笔记本需要多少钱？
5、Euclid高中在2002年有60名学生参加AMC10，在2003年有66名，2004年有70名，2005年有76名，2006年有78名，2007年有85名。

在相邻的哪两年中人数增长的百分率最大？
6、去年，John Q. Public先生继承了一笔遗产。

他需要支付20%的联邦税和剩下的部分中10%的州税。

他需要支付美元的总税费。

这笔遗产有多少美元？
答案：(C) 美元
有多少个正整数对(m, n)，满足m > n且它们的平方差是96？。

amc8历年真题答案解析

amc8历年真题答案解析【Amc8历年真题答案解析】Amc8，全名为American Mathematics Competition 8，是一项面向美国中学生的数学竞赛。

每年11月份举行的Amc8竞赛，旨在通过一系列挑战性的数学问题，培养学生的数学思维能力、创造力和解决问题的能力。

以下是Amc8历年真题的答案解析。

一、2000年真题：1. 答案：C解析：问题要求选择一个在100和999之间的正整数，满足各个数字的平方和等于该数本身。

首先明确数字的范围是100到999，因此只需要考虑三位数。

假设一个三位数为$abc$，那么它满足条件$a^2+b^2+c^2=abc$。

考虑到$a,b,c$的范围在1到9之间，可以通过穷举法逐个尝试。

当$a=2$时，$b=5$和$c=1$，满足条件$2^2+5^2+1^2=251$，因此答案为C。

2. 答案：D解析：问题给出了一个小测试，Tom回答7道题目，其中4道题回答正确。

考虑到每道问题只有两个选择（正确或错误），因此回答7道题目的所有可能性为$2^7=128$。

在这些可能性中，只有一种情况Tom回答了4道题目正确，因此答案为D。

3. 答案：A解析：问题给出了一个图形，包含了一系列长方形。

我们需要计算图形中所有（不重叠）长方形的个数。

首先考虑长方形最小边长为1的情况，可以在图形中找到4个这样的长方形。

然后考虑最小边长为2的情况，可以找到3个长方形。

以此类推，最小边长为3、4、5的情况下，分别可以找到2个、1个、0个长方形。

因此，总共可以找到的长方形个数为$4+3+2+1=10$，答案为A。

4. 答案：C解析：问题给出了一个5乘5的正方形网格，要求从左上角走到右下角，只能向右或向下移动，且不能重复经过同一格。

我们需要计算从左上角到右下角有几条路径。

可以发现，从左上角到右下角，一共需要移动4次向右，4次向下。

因此可以将问题转化为在8个移动中选择4个向右，即$C_8^4=\frac{8!}{4!4!}=70$，答案为C。

AMC-2000年

AMC 2000年中文试题与答案解析1.安娜阿姨42岁。

凯特琳比布里安娜小5岁，布里安娜的年龄是安娜阿姨的一半。

凯特琳多大年龄？答案与解析：B ，布里安娜的年龄为212142=⨯，凯特琳比布里安娜小5岁，因此凯特琳16521=-岁。

2.下列哪一个数字小于其倒数？答案与解析：A ，212--的倒数为，221->-，其他选项中，-1和1的倒数等于其自身，2的倒数为21，小于2，0没有倒数。

3.35和2 之间有多少个整数？(A)2(B)3(C)4(D)5(E)不确定答案与解析：D ，67.135≈，28.614.322=⨯≈π，因此，这两个数之间有5个整数，分别为2、3、4、5、6。

4.在1960年，卡林市只有5%的工作的成年人在家工作。

到1970年，在家工作的人数增加到了8%。

1980年，大约有15%的人在家工作，1990年有30%的人在家工作。

下列哪张图表最能说明这一变化过程？答案与解析：E ，显然，只有图E 满足题目叙述。

5.林肯高中的每一位校长都只有一个3年的任期。

在8年的时间里，这所学校最多可以有多少名校长？答案与解析：C ，下图为每位校长的任职的示意图，8年时间中最多可以有4名校长。

6.图形ABCD 是一个正方形。

在这个正方形内有三个较小的正方形，边长如图所示。

阴影L 形区域的面积为答案与解析：A ，三个较小的正方形边长为1和3，大正方形ABCD 的边长为5131=++，因此两块L 形区域的面积之和为14921253215222=-⨯-=-⨯-，一块L 形区域的面积为7214=÷。

7.集合中的三个不同数字的乘积的最小可能值是多少？答案与解析：B ，三个不同数字的乘积可能是正数，也可能是负数，要使之最小，则为负数。

经过尝试，最小的乘积为：()280758-=⨯⨯-。

8.如图所示，三个骰子堆叠在一起。

18个面中有7个面是可见的，剩下11个面隐藏在背面、底部或中间。

在此视图中不可见的面上的点数之和是多少？答案与解析：D ，每个骰子的六个面上的点数之和为()212661654321=÷⨯+=+++++，因此，3个骰子的所有点数之和是63321=⨯。

AMC10的真题答案及中文翻译

AMC10的真题及中文翻译1、One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?(A) 2 (B) 5 (C) 10 (D) 15 (E) 20中文：一张展览票全价为20美元。

Susan用优惠券买4张票打七五折。

Pam用优惠券买5张票打七折。

Pam比Susan多花了多少美元？2、An aquarium has a rectangular base that measures 100cm by 40cm and has a height of 50cm. It is filled with water to a height of 40cm. A brick with a rectangular base that measures 40cm by 20cm and a height of 10cm is placed in the aquarium. By how many centimeters does that water rise?(A) 0.5 (B) 1 (C) 1.5 (D) 2 (E)2.5中文：一个养鱼缸有100cm×40cm的底，高为50cm。

它装满水到40cm的高度。

把一个底为40cm×20cm，高为10cm的砖块放在这个养鱼缸里。

鱼缸里的水上升了多少厘米？3、The larger of two consecutive odd integers is three times the smaller. What is their sum?(A) 4 (B) 8 (C) 12 (D) 16 (E) 20中文：2个连续的奇整数中较大的数是较小的数的3倍。

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(A) $2.53 (B) $5.06 (C) $6.24 (D) $7.42 (E) $8.77
5. Points M and N are the midpoints of sides P A and P B of P AB. As P moves along a line that is parallel to side AB, how many of the four quantities listed below change?
(A) 0 (B) 4 (C) 6 (D) 7 (E) 9
First AMC ....... 10
2000
3
7. In rectangle ABCD, AD = 1, P is on AB, and DB and DP trisect ADC. What is the perimeter of BDP ?
14. Mrs. Walter gave an exam in a mathematics class of ﬁve students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71, 76, 80, 82, and 91. What was the last score Mrs. Walter entered?
(A) 21 (B) 60 (C) 119 (D) 180 (E) 231
First AMC ....... 10
2000
4
12. Figures 0, 1, 2, and 3 consist of 1, 5, 13, and 25 nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in ﬁgure 100?
(A) 10401 (B) 19801 (C) 20201 (D) 39801 (E) 40801
ﬁgure 0
ﬁgure 1
ﬁgure 2
ﬁgure 3
13. There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?
(a) the length of the segment M N (b) the perimeter of P AB (c) the area of P AB (d) the area of trapezoid ABN M
←
P M N .................................................................................
(C) There are as many freshmen as sophomores.
(D) There are twice as many freshmen as sophomores.
(E) There are ﬁve times as many freshmen as sophomores.
(A) 40 (B) 50 (C) 55 (D) 60 (E) 75
4. Chandra pays an on-line service provider a ﬁxed monthly fee plus an hourly charge for connect time. Her December bill was $12.48, but in January her bill was $17.54 because she used twice as much connect time as in December. What is the ﬁxed monthly fee?
(A) 71 (B) 76 (C) 80 (D) 82 (E) 91
15. Two non-zero real numbers, a and b, satisfy ab = a − b. Which of the following ab
is a possible value of + − ab? ba
First AMC ....... 10
2000
2
1. In the year 2001, the United States will host the International Mathematical Olympiad. Let I, M, and O be distinct positive integers such that the product I · M · O = 2001. What is the largest possible value of the sum I + M + O?
1
1
1
(A) −2
(B) − 2
(C) 3
(D) 2
(E) 2
First AMC ....... 10
2000
5
16. The diagram shows 28 lattice points, each one unit from its nearest neighbors.
Segment AB meets segment CD at E. Find the length of segment AE.
→
A
B
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
6. The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, . . . starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
(A) 23 (B) 55 (C) 99 (D) 111 (E) 671
2. 2000(20002000) = (A) 20002001 (B) 40002000 (C) 20004000 (D) 4, 000, 0002000 (E) 20004,000,000
3. Each day, Jenny ate 20% of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, 32 remained. How many jellybeans were in the jar originally?
(A) 2 (B) 4 (C) 6 (D) 8 (E) 10
11. Two diﬀerent prime numbers between 4 and 18 are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
17. Boris has an incredible coin changing machine. When he puts in a quarter, it returns ﬁve nickels; when he puts in a nickel, it returns ﬁve pennies; and when he puts in a penny, it returns ﬁve quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?
AP
B
D•• • ••C .........................................................................................................................................................
√
√
√
√
√
(A) 4 5/3 (B) 5 5/3 (C) 12 5/7 (D) 2 5 (E) 5 65/9
A• D• E• •C •B ••• •••• ••• •••• ••• •••• ••• .............................................................................................................................................................................................................................................................................................................