单项选择(2020年全国Ⅲ(文)

若(1 + i) = 1 − i, 则 z =【 】

A、1-i

B、1+i

C、-i

D、i

答案解析

D

讨论

求所有不超过100的正整数k,使得存在整数n,满足:k|(3n6+26n4+33n2+1)

设有理数r=p/q∈(0,1),其中p,q为互素的正整数,且pq整除3600.这样的有理数r的个数为________.

给定正整数k(k≥2)与k个非零实数a1,a2,⋯,ak.证明:至多有有限个k元整数组(n1,n2,⋯,nk),满足n1,n2,⋯,nk互不相同,且a1∙n1 !+a2∙n2 !+⋯+ak∙nk !=0.

设整数n≥4.证明:若n整除2n-2,则(2n-2)/n是合数.

In the sequence 7,76,769,7692,76923,769230,… ,the nth term is given by the first n digits after the decimal point in the expansion of 10/13=0.7692307692⋯.Prove that of the first 60 terms of the sequence, at least 49 have three or more prime factors (repeated prime factors are allowed; for example, 76=2×2×19 has three prime factors).【译】在10/13=0.7692307692⋯的十进制表示中,由小数点后的前n位数构成数列:7,76,769,7692,76923,769230,… ,求证:在该数列的前60项中,至少有49项有三个或以上的素因子(包含重复的素因子,例如76=2×2×19有三个素因子).

Let m<n be positive integers. Start with n piles, each of m objects. Repeatedly carry out the following operation: choose two piles and remove n objects in total from the two piles. For which (m ,n) is it possible to empty all the piles?【译】设正整数m<n.起初一共有n 堆石子,每堆有 m块石子. 重复执行以下操作: 选择两堆石子,从这两堆中移除共n 块石子.问:对于怎样的 (m , n),可以移除所有石子?

Consider an odd prime p and a positive integer N<50p. Let a1,a2,⋯,aN be a list of positive integers less than p such that any specific value occurs at most 51/100 N times and a1,a2,⋯,aN is not divisible by p. Prove that there exists a permutation b1,b2,⋯,bN of the a_i such that, for all k=1,2,⋯,N, the sum b1+b2+⋯+bk is not divisible by p.【译】已知奇素数p和正整数N<50p.设a1,a2,⋯,aN是一些小于p的正整数,同一数值至多出现51/100 N次,且a1+a2+⋯+aN不能被p整除.证明:存在a_i的一个排列:b1,b2,⋯,bN,使得对任意的k=1,2,⋯,N,都有b1+b2+⋯+bk不能被p整除.

Fix integers a and b greater than 1. For any positive integer n, let rn be the (non-negative) remainder that bn leaves upon division by an. Assume there exists a positive integer N such that rn<2n/n for all integers n≥N.Prove that a divides b.给定大于1的整数a和b.对任意的正整数n,记rn为bn除以an的非负余数.若存在正整数N,使得对任意的n≥N,都有rn<2n/n.证明:a整除b.

(2+2i)(1-2i)=【 】

已知z=1-2i,且z+az ̄+b=0,其中a,b为实数,则【 】