2026希望杯国际数学邀请赛巅峰对决IHC6-TT团队思考赛六年级数学试题

International Hope Cup Mathematics Invitational 2026 Global Peak Challenge (IHC 6) 2026希望杯国际数学邀请赛巅峰对决(HC6) Team Thinking Round团队思考赛 There are 10 questions in total,each worth 10 points,for a total of 100 points. Time limit:45 minutes. Discussion is allowed,each team answers one test paper,and only one answer sheet is submitted. 共10道题，每题10分，共100分。 限时：45分钟。 允许讨论，每个队答一份试卷，只提交一份答题卡。 TT-1.David discovers an interesting number pattern.He first writes a ten-digit number consisting of ten 8s:8888888888.Then he performs the following operation:erase the last digit of the number,multiply the remaining number by 2, and then add the digit that was just erased to obtain a new number.He repeatedly applies this operation to the new number.Eventually,a one-digit number is obtained.What is this one-digit number? David发现一个有趣的数字变化规律。他先写出由10个数字8组成的十 位数：8888888888。然后进行如下操作：擦去这个数的最后一位数字，将剩 下的数乘2，再加上刚才擦掉的数字，得到新数。不断对新数重复该操作， 最终会得到一个一位数，这个一位数是多少？ TT-2.Natural numbers a and b satisfy a+5=b.The sum of the digits of a is a multiple of 7,and the sum of the digits of b is also a multiple of 7.What is the smallest possible value of the natural number a? 自然数a,b满足a叶5=b,且a的数字和是7的倍数，b的数字和也是7的倍 数，满足条件的最小自然数是多少？ International Hope Cup Mathematics Invitational TT-3.There are 12 identical-looking balls.Among them,11 are genuine balls with equal weight,and the remaining 1 is a defective ball with an unknown weight The defective ball may be heavier or lighter than the genuine ones.Using only a balance scale without any weights,what is the minimum number of weighings required,in the worst-case scenario,to identify the defective ball and determine whether it is heavier or lighter? 现有12个外观完全相同的球，其中11个为重量相等的标准正品，剩余1个 为次品，次品重量未知，可能比正品重也可能比正品轻。仅使用一台无砝码 天平，要找出次品并判断次品偏重或偏轻，最坏的情况下最少需要称量几 次？ TT-4.Consider the sequence 1,1,2,1,2,4,1,2,4,8,1,2,4,8,16,...If the sum of the first Nterms of this sequence(N-100)is a power of 2,that is,2 where k is an integer,what is the minimum possible value of N? 数列1,1,2,1,2,4,1,2,4,8,1,2,4,8,16，，若该数列前N (>100)项的和为2(k为整数)，则N的最小值是多少？ TT-5.A set of chess pieces is stacked into a staircase shape.The top layer contains 1 piece,and each layer below contains 1 more piece than the layer above it.A total of n layers are stacked.If the total number of chess pieces minus 1 is a prime number,such a stack is called a "prime staircase."What are all the positive integers n that satisfy this condition? 将棋子堆成阶梯形，最上层放1枚棋子，向下每层的棋子数量比上一层多1 枚，共堆n层。若棋子总数减1的结果为质数，就称这样的堆法为“质数阶 梯”。所有符合条件的正整数n有哪些？ 2 Mathematics Ivitational International Hope Cup TT-6.There are 10 distinct natural numbers.The sum of the digits of each number is the same,and the sum of these 10 numbers is 2026.What is the greatest possible value of the largest number among these 10 numbers? 有10个互不相同的自然数，每个数的各位数字和相等，且这10个数的和为 2026。那么这10个数中，最大数所能取到的最大值是多少？ TT-7.What is the maximum number of consecutive positive integers such that each of them is divisible by at least one of the numbers 2,3,5,7,and 11? 最多能找到多少个连续正整数，它们中的每一个数都能被2、3、5、7、11 中至少一个数整除？ TT-8.There are two piles of wooden blocks containing 2026 blocks and 810 blocks, respectively.William and Frank take turns removing blocks.On each turn,a player may only take blocks from the pile with the larger number of blocks,and the number of blocks removed must be an integer multiple of the number of blocks in the other pile.The player who first empties either pile wins.If William goes first and both players use optimal strategies,how many blocks should William remove from the pile containing 2026 blocks on his first move to guarantee a win? 两堆木块数量分别为2026块和810块。William和Frank两人轮流取木块， 每次只能从数量较多的一堆中取木块，取走的块数必须是另一堆木块数量的 整数倍，率先取空任意一堆的人获胜。若William先取且双方均采用最优策 略，为确保必胜，W1liam第一次应从2026块的一堆中取走多少块？ ational Hope Cup internationat Hope Cup International Hope Cup Mathematics Invitational TT-9.There are 52 points arranged in a straight line,numbered from 1 to 52,with a distance of 1 between any two adjacent points.Alex starts at point 1.Each time, he may jump to any other point.After a total of 52 jumps,he returns to point 1, and every point has been visited at least once during the journey.What is the maximum possible total distance of all his jumps? 一条直线上依次排列着间距为1的52个点，编号1至52。A1ex从1号点出 发，每次可任意跳到其他点，总共跳跃52次后回到1号点，途中所有点都 到达过。他全程跳跃的总路程最大是多少？ TT-10.A positive integer that can be written as the product of two prime numbers is called a semiprime.If all the digits of a semiprime are even,and after removing any one digit from it,the remaining digits (in their original order)still form a semiprime,then it is called a perfect semiprime.For example,after removing any one digit from 2026,the resulting numbers 202,206,226,and 26 are all semiprimes.Therefore,2026 is a perfect semiprime.How many perfect semiprimes are there that are less than 2026? 一个正整数如果可以写成两个质数的乘积，就称它为半质数。如果一个半质 数的所有数字均为偶数，并且任意除去它某个数位上的数字，余下的数字按 顺序组成的数仍为半质数，就称它为完美半质数。例如2026除去某位数字 后得到202、206、226、26均为半质数，因此2026就是一个完美半质数。 那么小于2026的完美半质数共有多少个？ hational Hope Cup International Hope Cup 4 International Hope Cup Mathematics Invitational Answer (答案) Question TT-1 TT-2 TT-3 TT-4 TT-5 TT-6 tt-7 TT-8 TT-9 TT-10 题目 Answer 8 2999999 3 440 2,3 1531 13 810 1352 5 答案 International Hope Cup International Hope Cup Hope Cup International Hope Cup International Hope c nternational Hope Cup International Hope Cup