AMC10美国数学竞赛讲义

1、精选优质文档-倾情为你奉上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 7173 79 83 89 91 2：Perfect number:Let P is the prime number.if is also the prime number. then is the perfect number. For example:6,28,496. 3: Let is three digital integer .if Then the num

2、ber is called Daffodils number. There are only four numbers: 153 370 371 407 Let is four digital integer .if Then the number is called Roses number. There are only three numbers: 1634 8208 94744：The Fundamental Theorem of Arithmetic Every natural number n can be written as a product of primes unique

3、ly up to order. n=i=1kpiri5：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. For any integers a and b, we have gcd(a, b) = gcd(b, a) = gcd(±a, 

4、7;b) = gcd(a, b a) = gcd(a, b + a). For example: gcd(150, 60) = gcd(60, 30) = gcd(30, 0) = 30 (2)Least common multiple:Let lcm(a,b)=mindZ: a | d and b | d . (3)We have that: ab= gcd(a, b) lcm(a,b)7：Congruence modulo n If ,then we call a congruence b modulo m and we rewrite . (1)Assume a, b, c, d, m

5、,kZ (k>0, m0).If ab mod m, cd mod m then we have , , (2) The equation ax b (mod m) has a solution if and only if gcd(a, m) divides b. 8：How to find the unit digit of some special integers(1)How many zero at the end of For example, when, Let N be the number zero at the end of then (2) Find the uni

6、t digit. For example, when9：Palindrome, such as 83438, is a number that remains the same when its digits are reversed. There are some number not only palindrome but 112=121，222=484，114=14641(1)Some special palindrome that is also palindrome. For example :(2)How to create a palindrome? Almost integer

7、 plus the number of its reversed digits and repeat it again and again. Then we get a palindrome. For example: But whether any integer has this Property has yet to prove(3) The palindrome equation means that equation from left to right and right to left it all set up. For example： Let and are two dig

8、ital and three digital integers. If the digits satisfy the , then .10: Features of an integer divisible by some prime number If n is even，then 2|n 一个整数的所有位数上的数字之和是3（或者9）的倍数，则被3（或者9）整除 一个整数的尾数是零， 则被5整除 一个整数的后三位与截取后三位的数值的差被7、11、13整除，则被7、11、13整除 一个整数的最后两位数被4整除，则被4整除 一个整数的最后三位数被8整除，则被8整除 一个整数的奇数位之和与偶数位之

9、和的差被11整除，则被11整除 11. The number Theoretic functions If (1) (2) (3) For example: Exercise1. The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number? (A) 4 (B) 5 (C) 6 (D) 7(E) 83. For the positive integer n, let <n> denote the sum of all the positive divisors

10、 of n with the exception of n itself. For example, <4>=1+2=3 and <12>=1+2+3+4+6=16. What is <<<6>>>?(A) 6(B) 12(C) 24(D) 32(E) 368. What is the sum of all integer solutions to? (A) 10(B) 12(C) 15(D) 19(E) 510 How many ordered pairs of positive integers (M,N) satisfy the

11、 equation (A) 6(B) 7(C) 8(D) 9(E) 101. Let and be relatively prime integers with and. What is? (A) 1(B) 2(C) 3(D) 4(E) 515The figures and shown are the first in a sequence of figures. For, is constructed from by surrounding it with a square and placing one more diamond on each side of the new square

12、 than had on each side of its outside square. For example, figure has 13 diamonds. How many diamonds are there in figure? 18. Positive integers a, b, and c are randomly and independently selected with replacement from the set 1, 2, 3, 2010. What is the probability that is divisible by 3?(A) (B) (C)

13、(D) (E) 24. Let and be positive integers with such that and. What is? (A) 249(B) 250(C) 251(D) 252(E)253 5. In multiplying two positive integers a and b, Ron reversed the digits of the two-digit number a. His erroneous product was 161. What is the correct value of the product of a and b?(A) 116(B) 1

14、61(C) 204(D) 214 (E) 22423. What is the hundreds digit of?(A) 1(B) 4(C) 5(D) 6 (E) 99. A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers x and x+32 are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x?(A)

15、20(B) 21(C) 22(D) 23(E) 2421. The polynomial has three positive integer zeros. What is the smallest possible value of a?(A) 78(B) 88(C) 98(D) 108 (E) 11824. The number obtained from the last two nonzero digits of 90! Is equal to n. What is n?(A) 12(B) 32(C) 48(D) 52 (E) 6825. Jim starts with a posit

16、ive integer n and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with n=55, then his sequence contain 5 numbers:5555-72=66-22=22-12=11-12=0

17、Let N be the smallest number for which Jims sequence has 8 numbers. What is the units digit of N?(A) 1(B) 3(C) 5(D) 7 (E) 921What is the remainder when is divided by 8? (A) 0(B) 1(C) 2(D) 4(E) 65What is the sum of the digits of the square of? (A) 18(B) 27(C) 45(D) 63(E) 8125For, let, where there are

18、 zeros between the 1and the 6. Let be the number of factors of 2 in the prime factorization of. What is the maximum value of? (A) 6(B) 7(C) 8(D) 9(E) 1024. Let. What is the units digit of?(A) 0(B) 1(C) 4(D) 6 (E) 8AMC about algebraic problems一、Linear relations(1) Slope y-intercept form: (is the slop

19、e, is the y-intercept)(2)Standard form: (3)Slope and one point (4) Two points (5)x,y-intercept form: 二、the relations of the two lines (1) (1) 三、Special multiplication rules: 四、quadratic equations and PolynomialThe quadratic equations has two roots then we hasMore generally, if the polynomial has roo

20、ts ,then we have:开方的开方、估计开方数的大小绝对值方程Arithmetic SequenceIf n=2k, then we have If n=2k+1, then we have Geometric sequenceSome special sequence 1, 1, 2, 3, 5, 8, 9,99,999,9999，1，11，111，1111，Exercise4 .When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul

21、does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over? 7 For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chip

22、munk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide? 21. Four distinct points are arranged on a plane so that the segments conn

23、ecting them have lengths, and . What is the ratio of to? 6. The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?8. In a bag of marbles, of the marbles are blue and the rest are red. If

24、 the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?13. An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numb

25、ers, and then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?16. Three runners start runni

26、ng simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?2

27、4. Let and be positive integers with such that and. What is? (A) 249(B) 250(C) 251(D) 252(E) 253 1. What is?(A) -1(B) (C) (D) (E) 10. Consider the set of numbers 1, 10, 102, 1031010. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integ

28、er?(A) 1(B) 9(C) 10(D) 11 (E) 10119. What is the product of all the roots of the equation?(A) -64(B) -24(C) -9(D) 24 (E) 5764. Let X and Y be the following sums of arithmetic sequences:X= 10 + 12 + 14 + + 100.Y= 12 + 14 + 16 + + 102.What is the value of?(A) 92(B) 98(C) 100(D) 102(E) 1127. Which of t

29、he following equations does NOT have a solution?(A) (B) (C) (D) (E) 16. Which of the following in equal to?(A) (B) (C) (D) (E) 13. What is the sum of all the solutions of? (A) 32(B) 60(C) 92(D) 120 (E) 12414. The average of the numbers 1, 2, 3 98, 99, and x is 100x. What is x?(A) (B) (C) (D) (E) 11.

30、 The length of the interval of solutions of the inequality is 10. What is b-a?(A) 6(B) 10(C) 15(D) 20 (E) 3013. Angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total trip time o

31、f 3 hours including the stop. Which equation could be used to solve for the time t in hours that she drove before her stop?(A) (B) (C) (D) (E) 21. The polynomial has three positive integer zeros. What is the smallest possible value of a?(A) 78(B) 88(C) 98(D) 108 (E) 11815When a bucket is two-thirds

32、full of water, the bucket and water weigh kilograms. When the bucket is one-half full of water the total weight is kilograms. In terms of and, what is the total weight in kilograms when the bucket is full of water? 13Suppose that and . Which of the following is equal to for every pair of integers? 1

33、6Let, and be real numbers with , , and . What is the sum of all possible values of ? 5. Which of the following is equal to the product?(A) 251(B) 502(C) 1004(D) 2008 (E) 40167. The fraction simplifies to which of the following?(A) 1(B) 9/4(C) 3(D) 9/2 (E) 913. Doug can paint a room in 5 hours. Dave

34、can paint the same room in 7 hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let t 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 t?(A) (B) (C)(D) (E) 15. Yesterd

35、ay Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?(A) 120(B) 130(C) 140(D) 150 (E) 160AMC中的几何问

36、题一、三角形有关知识点1.三角形的简单性质与几个面积公式三角形任何两边之和大于第三边；三角形任何两边之差小于第三边；三角形三个内角的和等于180°；三角形三个外角的和等于360°；三角形一个外角等于和它不相邻的两个内角的和；三角形一个外角大于任何一个和它不相邻的内角。 设三角形ABC的三个角A，B，C对应的边是a,b,c,以A为顶点的高为h。则的三角形的面积公式有：；,其中r是内切圆半径； 2.直角三角形的相关定理（勾股、射影）直角三角形的识别：有一个角等于90°的三角形是直角三角形；有两个角互余的三角形是直角三角形；勾股定理定理：两个直角边的平方和等于斜边的平方

37、勾股定理逆定理：三角形两边的平方和等于第三边的平方，那么这个三角形是直角三角形。直角三角形的性质：直角三角形的两个锐角互余；直角三角形斜边上的中线等于斜边的一半；射影定理：如图直角三角形中过直角点向斜边做垂线则有3.正三角形的数据：等边三角形ABC如上图，分别作ABC的内切圆和外接圆，设等边三角形的边长为a,则4.其它特殊三角形： 等腰三角形性质：等边对等角；等腰三角形的顶角平分线、底边上的中线、底边上的高互相重合；等腰三角形是轴对称图形，底边的中垂线是它的对称轴；5.三角形的四心：三角形的三条中线交于一点，这个点叫做三角形的重心，重心将每一条中线分成1:2；三角形三边的垂直平分线交于一点，这

38、个点叫做三角形的外心，三角形的外心到各顶点的距离相等；三角形的三条角平分线交于一点，这个点叫做三角形的内心，三角形的内心到三边的距离相等；三角形三条垂线交于一点，这个点叫做三角形的垂心。6.三角形全等与相似：二、正六边形ABCDEF的性质，设AB=a则正六边形ABCDEF被三条对角线分成了六个全等的等边三角形.三、正四面体数据如上图，设正四面体ABCD的棱长为a,则有：1.正四面体是由四个全等正围成的空间封闭图形。 它有4个面，6条棱，4个顶点。正四面体是最简单的。 正四面体的重心、四条高的交点、外接球、内切球球心共点，此点称为中心。 正四面体有一个在其内部的内切球和一个外切球 正四面体有四条

39、三重旋转对称轴，六个对称面。 正四面体可与正八面体填满空间，在任意顶点周围有八个正四面体和六个正八面体。 2.相关数据当正四面体的棱长为a时，一些数据如下： 高：。中心O把高分为1:3两部分。 表面积： 体积：对棱中点的连线段的长： 外接球半径：， 内切球半径：， 两邻面夹角： 正四面体的对棱相等。具有该性质的四面体符合以下条件： 1四面体为对棱相等的四面体当且仅当四面体每对对棱的中点的连线垂直于这两条棱。 2四面体为对棱相等的四面体当且仅当四面体每对对棱中点的三条连线相互垂直。 3四面体为对棱相等的四面体当且仅当四条中线相等。 四、正方体相关数据：1.如图，设正方体的棱长为a,则面对角线为，

40、体对角线为，体对角线不仅与截面、垂直，而且被截面与截面分成三等分。2.正方体的八个顶点中的每四个面对角线的顶点构成了一个棱长为的正四面体。即与是一个棱长为的正四面体。这两个正四面体的相交部分是一个正八面体（恰好是正方体六个面的中心的连线）。3.正方体六个面的中心的连线构成一个棱长为的正八面体，体积是正方体的4.正方体在各个方向的投影的面积最大为5截面的性质：正方体的截面中会出现（见下图）：三角形、正方形、梯形、菱形、矩形、平行四边形、五边形、六边形、正六边形。其中三角形还分为锐角三角型、等边、等腰三角形。梯形分位非等腰梯形和等腰梯形。不可能出现：钝角三角形、直角三角形、直角梯形、正五边形、七边

41、形或更多边形。6.最大截面：最大截面四边形，如图所示的矩形：五、正八边形与正八面体：正八边形：设正八边形的棱长为a,面积是为，四边形、是正方形。正八边形有20条对角线(更一般的凸边形有条对角线，内部有49个交点(这个推广还没有统一的结论，是一个较为困难的问题)。正八面体：和都是正方形，内切球的半径，外接球半径，体积为六、圆和球：切割线、切割线定理（1）相交弦定理：圆内两弦相交，交点分得的两条线段的乘积相等。即：在中，弦、相交于点， （2）推论：如果弦与直径垂直相交，那么弦的一半是它分直径所成的两条线段的比例中项。即：在中，直径， （3）切割线定理：从圆外一点引圆的切线和割线，切线长是这点到割线

42、与圆交点的两条线段长的比例中项。即：在中，是切线，是割线 球的相关公式：球的体积、表面积公式：，Exercise2 A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? 3 The point in the xy-plane with coordinates (1000, 2012) is reflected across the line

43、 y=2000. What are the coordinates of the reflected point? 12 Point B is due east of point A. Point C is due north of point B. The distance between points A and C is , and = 45 degrees. Point D is 20 meters due North of point C. The distance AD is between which two integers? 14 Two equilateral triang

44、les are contained in square whose side length is . The bases of these triangles are the opposite side of the square, and their intersection is a rhombus. What is the area of the rhombus? 16 Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded

45、 by them, as shown in the figure? 17 Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smal

46、ler cone to that of the larger? 23 A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the

47、cube is placed on a table with the cut surface face down. What is the height of this object? 24The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids ar

48、e horizontal. In an arch made with trapezoids, let be the angle measure in degrees of the larger interior angle of the trapezoid. What is ? 10. Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What

49、 is the degree measure of the smallest possible sector angle? 11. Externally tangent circles with centers at points A and B have radii of lengths 5 and 3, respectively. A line externally tangent to both circles intersects ray AB at point C. What is BC? 15. Three unit squares and two line segments co

50、nnecting two pairs of vertices are shown. What is the area of? 21. Let points =, =, =, and =. Points, and are midpoints of line segments and respectively. What is the area of? 9. The area of EBD is one third of the area of 3-4-5 ABC. Segment DE is perpendicular to segment AB. What is BD?(A) (B) (C)

51、(D) (E) 16. A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?(A) (B) (C) (D) (E) 17. In the given circle, the diameter EB is paral

52、lel to DC, and AB is parallel to ED. The angles AEB and ABE are in the ratio 4:5. What is the degree measure of angle BCD?(A) 120(B) 125(C) 130(D) 135 (E) 14020. Rhombus ABCD has side length 2 and B=120°. Region R consists of all points inside the rhombus that are closer to vertex B than any of

53、 the other three vertices. What is the area of R?(A) (B) (C) (D) (E) 22. A pyramid has a square base with sides of length land 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

54、on the lateral faces of the pyramid. What is the volume of this cube?(A) (B) (C) (D) (E) 11. Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE=7·EB. What is the ratio of the area of EFGH to the area of ABCD?(A) (B) (C) (D) (E) 24. Two distinct regular tetrahedra h

55、ave all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?(A) (B) (C) (D) (E) 16. A square of side length 1 and a circle of radius share the same center. What is the area inside the circle, but outside the square?(A) (B) (C) (D) (E) 19. A circle with center O has area 156. Triangle ABC is equilateral, BC is a chord on the circle, , and point O is outside ABC. What is the side length of ABC?(A) (B) 64(C) (D) 12 (E) 1820. Two