填空题(2024年清华大学

令S=m²n/(2m(n2m+m2n)),则[100S]=________.

答案解析

200

【解析】

解答过程见word版

讨论

令S=(lnx)5/x²dx,则[S]=______.

令A,B,C,D,E,F是三阶实方阵,且=.已知A=,B=且C=A+B-I,则[|detF|]=______.

已知圆锥面x²+y²=z²/3,记沿该圆锥面从P(-√3,3,6)到Q(√3,0,3)的曲线长度的最小值为I,则[10I]=________.

对于一个实数x,令{x}=x-[x]. 记S=min⁡({x/8},{x/4}) dx,则[S]=______.

A polynomial P with integer coefficients is square-free if it is not expressible in the form P=Q² R, where Q and R are polynomials with integer coefficients and Q is not constant. For a positive integer n, let Pn be the set of polynomials of the form1+a1 x+a2 x²+⋯+an xnwith a1,a2,⋯,an∈{0,1}. Prove that there exists an integer N so that, for all integers n>N, more than 99% of the polynomials in Pn are square-free.【译】我们称整系数多项式P是无平方因子的,如果其不能表示为P=Q² R的形式,这里Q,R为整系数多项式且Q不为常数.对于正整数n,记Pn为如下 形式的多项式组成的集合:1+a1 x+a2 x²+⋯+an xn这里a1,a2,⋯,an∈{0,1}.证明:存在整数N,使得对任意的整数n≥N,Pn中超过99%的多项式都是无平方因子的.

Let BC be a fixed segment in the plane, and let A be a variable point in the plane not on the line BC. Distinct points X and Y are chosen on the rays (CA) ⃗ and (BA) ⃗, respectively, such that ∠CBX=∠YCB=∠BAC.Assume that the tangents to the circumcircle of ABC at B and C meet line XY at P and Q, respectively, such that the points X,P,Y, and Q are pairwise distinct and lie on the same side of BC. Let Ω1 be the circle through X and Y centred on BC. Similarly let Ω2 be the circle through Y and Q centred on BC. Prove that Ω1 and Ω2 intersect at two fixed points as A varies.【译】在同一平面内,BC为给定线段,动点A不在直线BC上. X和Y分别为射线(CA) ⃗,射线(BA) ⃗上不重合的两点,满足∠CBX=∠YCB=∠BAC.若三角形ABC外接圆在点B和C处的切线分别交直线XY于点P和点Q,点X,P,Y,Q不重合,且位于直线BC同侧.圆Ω1经过点X,P且圆心在BC上.类地,圆Ω2经过点Y,Q且圆心在BC上.证明:当点A运动时,圆Ω1和圆Ω2始终交于两定点.

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.

Given a positive integer n, a set S is n-admissible if①each element of S is an unordered triple of integers in {1,2,⋯,n},②|S|=n-2,and③for each 1≤k≤n-2 and each choice of k distinct A1,A2,⋯,Ak∈S,|A1∪A2∪⋯∪Ak |≥k+2Is it true that, for all n>3 and for each n-admissible set S, there exist pairwise distinct points P1,P2,⋯,Pn in the plane such that the angles of the triangle Pi Pj Pk are all less than 61° for any triple {i,j,k} in S?【译】给定正整数n,称集合S是n-可行,如果其满足以下条件:①S的每个元素都是{1,2,⋯,n}的三元子集;②|S|=n-2;③对任意的1≤k≤n-2和任意k个互不相同的A1,A2,⋯,Ak∈S,都有|A1∪A2∪⋯∪Ak |≥k+2判断以下命题是否为真:对所有n>3和所有的n-可行集合S,在平面内总存在n个互不相同的点P1,P2,⋯,Pn,使得对集合S中任意元素{i,j,k},三角形Pi Pj Pk的每个内角都小于61°.

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.

Let n be a positive integer. Initially, a bishop is placed in each square of the top row of a 2n×2n chessboard; those bishops are numbered from 1 to 2n ,from left to right. A jump is a simultaneous move made by all bishops such that the following conditions are satisfied:each bishop moves diagonally, in a straight line, some number of squares, andat the end of the jump, the bishops all stand in different squares of the same row.Find the total number of permutations σ of the numbers 1,2,⋯,2n with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order σ(1),σ(2),⋯,σ(2n ), from left to right.【译】设n是正整数.最开始在一个2n×2n的方格棋盘上的第一行的每个小方格内均放置一枚“象”,这些“象”从左到右依次编号:1,2,⋯,2n.定义一次“跳跃”操作为同时移动所有的“象”并满足如下条件:每一枚“象”可沿对角线方向移动任意方格;在这次“跳跃”操作结束时,所有的“象”恰在同一行的不同方格.求满足下列条件的数1,2,⋯,2n的排列σ的总个数:存在一系列的“跳跃”操作,使得结束时所有的“象”都在棋盘的最后一行,并且从左到右编号依次为:σ(1),σ(2),⋯,σ(2n ).

给定整数k≥2.求所有无穷正整数数列a1,a2,⋯,使得存在多项式P(x)=xk+ck-1 xk-1+⋯+c1 x+c0其中c0,c1,⋯,ck-1是非负整数,满足P(an )=an+1 an+2⋯an+k对任意正整数n成立.

已知数列{an}满足an+1=1/4 (an-6)³+6(n=1,2,3,⋯),则【 】

我国度量衡的发展有着悠久的历史,战国时期就已经出现了类似于砝码的、用来测量物林质量的“环权”,已知9枚环权的质量(单位:铢)从小到大构成项数为9的数列{an},该数列的前3项成等差数列,后7项成等比数列,且a1=1,a5=12,a9=192,则a7=______;数列{an}所有项的和为________.

已知数列{an },{bn}的项数均为m(m>2),且an,bn∈{1,2,⋯,m},{an },{bn}的前n项和分别为An,Bn,并规定A0=B0=0.对于k∈{0,1,2,⋯,m},定义rk=max⁡{i|Bi≤Ai,i∈{0,1,2,⋯,m}},其中maxM表示数集M中最大的数.(1)若a1=2,a2=1,a3=3,b1=1,b2=3,b3=3,求r0,r1,r2,r3的值;(2)若a1≥b1,2rj≤rj+1+rj-1,j=1,2,⋯,m-1,求rn;(3)证明:存在p,q,s,t∈{0,1,2,⋯,m},满足p>q,s>t,使得Ap+Bt=Aq+Bs.

等差数列{an}满足a2021=a20+a21=1,则a1的值为__________.

丘成桐女子赛数列极限

在公比为正数的等比数列{an}中,a2+a4=30,a4+a6=15/2,则a1的值为【 】

有三数原成等比级数,其和为9/2.若第一数以2/3乘之,第二数以2/3乘之,第三数以16/27乘之,则成等差级数,问原三数各几何?

求级数1/(1×3)+1/(3×5)+1/(5×7)+⋯ n项及无穷项之和.其第n项为1/(2n-1)(2n+1).

有相交之二直线 a 及 b,自 a 上之一点作 b 之垂线,复自其在 b 上之垂足向 a作垂线,更自第二个垂足作 b 之垂线,如此继续作成无数根垂线,设第一垂线之长为 7,第二垂线之长为 6,求此无数垂线长之和.