AMC10美國數(shù)學(xué)競賽講義

1、AMC中的數(shù)論問題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 number is called

2、 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 uniquely up to orde

3、r. 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, ±b) = gcd(a,

4、 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 ,kZ (k>0,

5、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 unit digit. For

6、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 plus the num

7、ber 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 digital and thre

8、e 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 一個整數(shù)的所有位數(shù)上的數(shù)字之和是3（或者9）的倍數(shù)，則被3（或者9）整除 一個整數(shù)的尾數(shù)是零， 則被5整除 一個整數(shù)的后三位與截取后三位的數(shù)值的差被7、11、13整除，則被7、11、13整除 一個整數(shù)的最后兩位數(shù)被4整除，則被4整除 一個整數(shù)的最后三位數(shù)被8整除，則被8整除 一個整數(shù)的奇數(shù)位之和與偶數(shù)位之和的差被11整除，則被11

9、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 of n with th

10、e 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 equation (A)

11、 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 than had on

12、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) (D) (E) 24. L

13、et 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) 161(C) 204(D)

14、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) 20(B) 21(C) 2

15、2(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 positive integer n

16、 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=0Let N be the

17、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 zeros betwee

18、n 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 slope, is the y-i

19、ntercept)(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 roots ,then we h

20、ave:開方的開方、估計開方數(shù)的大小絕對值方程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 does the same

21、 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 chipmunk hid 3 ac

22、orns 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 connecting them h

23、ave 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 the number o

24、f 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 numbers, and then

25、 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 running simultaneo

26、usly 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?24. Let and be

27、 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 integer?(A) 1(B) 9

28、(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 the following

29、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. The length o

30、f 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 of 3 hours inc

31、luding 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 full of water

32、, 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? 16Let, and be

33、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 can paint the

34、 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. Yesterday Han drove

35、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中的幾何問題一、三角形有關(guān)知識點1.

36、三角形的簡單性質(zhì)與幾個面積公式三角形任何兩邊之和大于第三邊；三角形任何兩邊之差小于第三邊；三角形三個內(nèi)角的和等于180°；三角形三個外角的和等于360°；三角形一個外角等于和它不相鄰的兩個內(nèi)角的和；三角形一個外角大于任何一個和它不相鄰的內(nèi)角。 設(shè)三角形ABC的三個角A，B，C對應(yīng)的邊是a,b,c,以A為頂點的高為h。則的三角形的面積公式有：；,其中r是內(nèi)切圓半徑； 2.直角三角形的相關(guān)定理（勾股、射影）直角三角形的識別：有一個角等于90°的三角形是直角三角形；有兩個角互余的三角形是直角三角形；勾股定理定理：兩個直角邊的平方和等于斜邊的平方勾股定理逆定理：三角形兩邊

37、的平方和等于第三邊的平方，那么這個三角形是直角三角形。直角三角形的性質(zhì)：直角三角形的兩個銳角互余；直角三角形斜邊上的中線等于斜邊的一半；射影定理：如圖直角三角形中過直角點向斜邊做垂線則有3.正三角形的數(shù)據(jù)：等邊三角形ABC如上圖，分別作ABC的內(nèi)切圓和外接圓，設(shè)等邊三角形的邊長為a,則4.其它特殊三角形： 等腰三角形性質(zhì)：等邊對等角；等腰三角形的頂角平分線、底邊上的中線、底邊上的高互相重合；等腰三角形是軸對稱圖形，底邊的中垂線是它的對稱軸；5.三角形的四心：三角形的三條中線交于一點，這個點叫做三角形的重心，重心將每一條中線分成1:2；三角形三邊的垂直平分線交于一點，這個點叫做三角形的外心，三角

38、形的外心到各頂點的距離相等；三角形的三條角平分線交于一點，這個點叫做三角形的內(nèi)心，三角形的內(nèi)心到三邊的距離相等；三角形三條垂線交于一點，這個點叫做三角形的垂心。6.三角形全等與相似：二、正六邊形ABCDEF的性質(zhì)，設(shè)AB=a則正六邊形ABCDEF被三條對角線分成了六個全等的等邊三角形.三、正四面體數(shù)據(jù)如上圖，設(shè)正四面體ABCD的棱長為a,則有：1.正四面體是由四個全等正三角形圍成的空間封閉圖形。 它有4個面，6條棱，4個頂點。正四面體是最簡單的正多面體。 正四面體的重心、四條高的交點、外接球、內(nèi)切球球心共點，此點稱為中心。 正四面體有一個在其內(nèi)部的內(nèi)切球和一個外切球 正四面體有四條三重旋轉(zhuǎn)對稱

39、軸，六個對稱面。 正四面體可與正八面體填滿空間，在任意頂點周圍有八個正四面體和六個正八面體。 2.相關(guān)數(shù)據(jù)當正四面體的棱長為a時，一些數(shù)據(jù)如下： 高：。中心O把高分為1:3兩部分。 表面積： 體積：對棱中點的連線段的長： 外接球半徑：， 內(nèi)切球半徑：， 兩鄰面夾角： 正四面體的對棱相等。具有該性質(zhì)的四面體符合以下條件： 1四面體為對棱相等的四面體當且僅當四面體每對對棱的中點的連線垂直于這兩條棱。 2四面體為對棱相等的四面體當且僅當四面體每對對棱中點的三條連線相互垂直。 3四面體為對棱相等的四面體當且僅當四條中線相等。 四、正方體相關(guān)數(shù)據(jù)：1.如圖，設(shè)正方體的棱長為a,則面對角線為，體對角線為，

40、體對角線不僅與截面、垂直，而且被截面與截面分成三等分。2.正方體的八個頂點中的每四個面對角線的頂點構(gòu)成了一個棱長為的正四面體。即與是一個棱長為的正四面體。這兩個正四面體的相交部分是一個正八面體（恰好是正方體六個面的中心的連線）。3.正方體六個面的中心的連線構(gòu)成一個棱長為的正八面體，體積是正方體的4.正方體在各個方向的投影的面積最大為5截面的性質(zhì)：正方體的截面中會出現(xiàn)（見下圖）：三角形、正方形、梯形、菱形、矩形、平行四邊形、五邊形、六邊形、正六邊形。其中三角形還分為銳角三角型、等邊、等腰三角形。梯形分位非等腰梯形和等腰梯形。不可能出現(xiàn)：鈍角三角形、直角三角形、直角梯形、正五邊形、七邊形或更多邊形

41、。6.最大截面：最大截面四邊形，如圖所示的矩形：五、正八邊形與正八面體：正八邊形：設(shè)正八邊形的棱長為a,面積是為，四邊形、是正方形。正八邊形有20條對角線(更一般的凸邊形有條對角線，內(nèi)部有49個交點(這個推廣還沒有統(tǒng)一的結(jié)論，是一個較為困難的問題)。正八面體：和都是正方形，內(nèi)切球的半徑，外接球半徑，體積為六、圓和球：切割線、切割線定理（1）相交弦定理：圓內(nèi)兩弦相交，交點分得的兩條線段的乘積相等。即：在中，弦、相交于點， （2）推論：如果弦與直徑垂直相交，那么弦的一半是它分直徑所成的兩條線段的比例中項。即：在中，直徑， （3）切割線定理：從圓外一點引圓的切線和割線，切線長是這點到割線與圓交點的兩

42、條線段長的比例中項。即：在中，是切線，是割線 球的相關(guān)公式：球的體積、表面積公式：，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 y=200

43、0. 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 triangles ar

44、e 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 by th

45、em, 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 smaller co

46、ne 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 cube i

47、s 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 are hori

48、zontal. 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 is th

49、e 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 connecti

50、ng 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) (D) (E

51、) 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 parallel to

52、 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 the o

53、ther 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 on the

54、 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 have al

55、l 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 c