高中数学-竞赛试题(罗马尼亚 2024年数论)

证明题（2024年罗马尼亚）

Fix integers a and b greater than 1. For any positive integer n, let r_n be the (non-negative) remainder that bⁿ leaves upon division by aⁿ. Assume there exists a positive integer N such that r_n<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r_n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r_n<2ⁿ/n.证明：a整除b.

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讨论

高中数学

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 ).

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?

Let ABC be an acute-angled triangle with AB > AC. Let P be the intersection of the tangents to the circumcircle of ABC at B and C. The line through the midpoints of line segments PB and PC meets lines AB and AC at X and Y respectively.Prove that the quadrilateral AXPY is cyclic.【译】在锐角三角形ABC中，AB>AC，△ABC的外接圆在点B和点C处的切线交于点P.一条同时过PB和PC中点的直线与AB,AC分别交于点X,Y.求证：A,X,P,Y四点共圆.

Find all functions f from the integers to the integers such that for all integers n：2f(f(n))=5f(n)-2n【译】求所有函数f:z→z，使得对任意整数n有：2f(f(n))=5f(n)-2n

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).

设S={z∈C||z|=1}.求所有函数f:S→S，使得f为连续单射,且对任意z1,z2∈S，有f(z1 z2 )=f(z1)f(z2).

复矩阵A与A的任意正整数次常相似.(1)证明:A的特征值为0或 1;(2)求A的若当标准型.

给定素数p和正整数 n(n≥2).A为n个p阶循环群的直和.问:至少需要几个A的真子群,才能使他们的并集能覆盖A？

数论

Determine all composite integers n>1 that satisfy the following property：if d1,d2,⋯,dk are all the positive divisors of n with 1=d1<d2<⋯<dk=n, then di divides di+1+di+2 for every 1≤i≤k-2.译文：设1=d1<d2<⋯<dk=n是合数n的全部正因数，若对任意1≤i≤k-2，有di |di+1+di+2，求n.

设x1,x2,⋯,x2023为两两不等的正实数，对任意一个n=1,2,⋯,2023，an=都是一个整数.证明：a2023≥3034.

使得n²+2023n为平方数的正整数n的最小值是__________.

已知a,b为正整数，a<b，且a,b互质.若关于x,y的不等式ax+by≤ab有且仅有2023组正整数解，则(a,b)=____________________(求出满足题意的所有可能数组).

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.

今有三数，其和为 37，积为 1440，且其中二数的积较第三数的三倍大 12.试求此三数.

一个分数的分子与分母之和为 38，其分子和分母都减去15，约分后得到1/3，则这个分数的分母与分子之差为【】

数系与复数

在复平面内，复数z对应的点的坐标是(-1,√3)，则z的共轭复数z ̅=【】

已知复数列{zn}满足：z1=√3/2,zn+1=zn ̅(1+zni)(n=1,2,⋯)其中i为虚单位.求z2021的值.

已知z=2-i，则z(z ̅+i)=【】

已知(1-i)2z=3+2i，则z=【】

设2(z+z ̅)+3(z - z ̅)=4+6i，则z=【】

设iz=4+3i，则z=【】

在复平面内，复数z满足(1-i)·z=2，则z=【】

z1=1+i,z2=2+3i，则z1+z2=__________.

设i是虚数单位，复数(9+2i)/(2+i)=__________.

已知a∈R,(1+ai)i=3+i，(i为虚单位)，则a=【】