AMC10美国数学竞赛讲义(共39页).docx

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 number 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 uniquely 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, 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)=min{d∈Z: 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 ,k∈Z (k>0, m≠0).If a≡b mod m, c≡d 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 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 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 digital 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 denote the sum of all the positive divisors 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 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) 515．The 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 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. 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) 161 (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) 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 positive integer n and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible int。