资源简介

2026 Global Peak Challenge (IHC A)
2026 希望杯国际数学邀请赛巅峰对决（IHC A）
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. Given the set S={1,2,3,…,20}. If the setM ,M S , and the sum of all
elements in M is a multiple of 5, find the number of possible sets M.
已知集合 S={1，2，3，…，20}。若集合M ，M S，且 M中所有元素
之和是 5的倍数，求集合 M的个数。
log3 x，0 x 27
TT-2. Let f (x)

x . Suppose that three distinct real numbers a, b,
6, x 27 9
and c satisfy f(a)=f(b)=f(c). Determine the range of possible values of abc.
log3 x， 0 x 27
设函数 f (x)

x 。若互不相等的实数 a，b，c满足 f(a)=f(b)
6， x 27 9
=f(c)，求 abc的取值范围。
1
TT-3. Definition: [x] represents the greatest integer less than or equal to x(the floor
6
function). Find the value of 5 3 .
6
定义：[x]表示不超过 x的最大整数，求 5 3 的值。
TT-4. In the triangle ABC, sin A cosB 3 , tan A sin B 3 , find the value of
4 2
sinA.
3
在△ABC中， sin A cosB 3 ， tan A sin B ，求 sinA的值。
4 2
TT-5. Given positive real numbers a1, a2, ···, a12 satisfying a1+a2+···+a12=14, find
the minimum value of 2 a21 2 a2 22 2 a12 .
已知正实数 a1，a2，···，a12满足 a1+a2+···+a12=14，
求 2 a2 2 21 2 a2 2 a12 的最小值。
TT-6. A sequence of positive integers {an} satisfies a1all sums ai+aj , where 1≤i, j≤7 and i≠j, contains exactly 11 different values. Given
that a1=1, find the maximum possible value of a7.
正整数数列{an}满足：a1不相同的数恰好有 11个。若 a1=1，求 a7的最大值。
TT-7. In the triangular pyramid P-ABC, the base ABC is a triangle with side lengths
6, 8, and 10. If the sum of the lengths of two lateral edges is 12, find the
maximum possible volume of the triangular pyramid P-ABC.
三棱锥 P-ABC中，底面 ABC是边长为 6，8，10的三角形。若某两条侧棱
长度的和为 12，求三棱锥 P-ABC体积的最大值。
2
TT-8. Let A1A2A3A4A5A6 be a convex hexagon. Find the maximum possible value of
sinA1+sinA2+···+sinA6,where∠Ai denotes the interior angle at vertex Ai(i=1,2,···).
已知 A1A2A3A4A5A6是一个凸六边形，求 sinA1+sinA2+···+sinA6的最大值。
TT-9. It is known that cos5θ=acosθ+bcos3θ+ccos5θ, where a, b, and c are constants.
Find the value of a2+b2+c2.
已知 cos5θ=acosθ+bcos3θ+ccos5θ，其中 a，b，c为常数，求 a2+b2+c2的值。
TT-10. In the FIFA World Cup group stage, each group consists of four teams
playing a single round-robin tournament. A win earns 3 points, a draw earns 1
point, and a loss earns 0 points. After all six matches have been played, how
many different point distributions are possible
世界杯小组赛采用 4支球队单循环赛制，胜一场积 3分，平一场积 1分，
负一场积 0分。全部 6场比赛结束后，可能出现多少种不同的积分组合？
3
Answer（答案）
Question
TT-1 TT-2 TT-3 TT-4 TT-5
题目
Answer
209727 (27,54) 3903 3 22
答案 2
Question
TT-6 TT-7 TT-8 TT-9 TT-10
题目
Answer
63
61 24 3 3 3 40
答案 128
4

展开更多......

收起↑