资源简介

2026 Global Peak Challenge(IHC 8)
2026 希望杯国际数学邀请赛巅峰对决（IHC 8）
Individual Thinking Round 个人思考赛
Total of 10 questions, each worth 10 points, for a total of 100 points.
Time limit: 30 minutes.
No discussion allowed. Each person answers one exam paper and submits one answer
sheet.
共 10道题，每题 10分，共 100分。
限时：30分钟。
不允许讨论，每人答一份试卷，各提交一份答题卡。
I-1. Given that and satisfy 1 a (b 1) 1 b 0 , find the value of a2026－
b2027.
已知 a，b满足 1 a (b 1) 1 b 0，求 a2026－b2027的值。
I-2. As shown in the figure, in right triangle ABC, the right-angled side AC = 30 and
BC = 40. A circle is drawn with the altitude CD of the hypotenuse as its diameter,
and a semicircle is drawn with the right-angled side AC as its diameter. Find the
area of the shaded region in the figure. (Take π as 3)
如图，在 Rt△ABC中，直角边 AC=30，BC=40。以斜边的高 CD为直径作圆，
以直角边 AC为直径作半圆，求图中阴影部分的面积。（π取 3）
1
I-3. In a Cartesian coordinate system, each vertex of a rectangle is an integer (i.e.,
both its x and y coordinates are integers), and its four sides are parallel to the
coordinate axes. If the x-coordinates of all four vertices of a rectangle satisfy 2 <
x < 7, and the y-coordinates satisfy 2 < y < 6, how many rectangles satisfy these
conditions
在平面直角坐标系中，长方形各顶点均为整点（即横、纵坐标均为整数），
且长方形四边分别平行于坐标轴。若长方形四个顶点的横坐标均满足 2 < x <
7，纵坐标均满足 2 < y < 6，则符合条件的长方形共有多少个？
I-4. In parallelogram ABCD, square ABEF is constructed outwards with AB as a side,
and square ADGH is constructed outwards with AD as a side. The area of square
ABEF is 4, and the area of square ADGH is 8. Connect CE, DH, and FH. Given
that points C, D, and H are collinear, find the area of trapezoid FECH.
在平行四边形 ABCD中，以 AB为边向外作正方形 ABEF，以 AD为边向外作
正方形 ADGH，其中正方形 ABEF面积为 4，正方形 ADGH面积为 8。连接
CE，DH，FH，已知 C，D，H三点共线，求梯形 FECH的面积。
I-5. Alice, Bill, and Charlie solved a total of 108 math problems, with each of them
solving 77. Problems that only one person could solve are called difficult
problems, and problems that all three could solve are called easy problems. How
many more easy problems were there than difficult problems
Alice、Bill和 Charlie三人合计解出 108道不同的数学题，每人各解出 77道。
仅一人解出的题目称为难题，三人都解出的题目称为容易题，则容易题比难
题多几道？
2
I-6. In the group stage of the World Cup, each group consists of four teams playing a
round-robin tournament. Each team plays three matches. A win earns 3 points, a
draw earns 1 point, and a loss earns 0 points. The top two teams in each group
advance to the next round. If teams have the same total points, their rankings are
determined according to additional tiebreaker rules, and the top two teams still
qualify. What is the highest number of points a team can earn while still having
the possibility of failing to advance
在世界杯小组赛中，每个小组有 4支球队进行单循环比赛，每支球队各踢 3
场，胜一场积 3分，平一场积 1分，负一场积 0分，小组排名前两名可以出
线；总积分相同时依据后续附加规则确定名次，依旧取前两名出线。那么一
支球队最高拿到多少分，仍然有无法出线的可能。
I-7. How many pairs of positive integers ( x, y ) satisfy the Diophantine equation
1 1 1

x y 2026
1 1 1
不定方程 x y 2026 有多少组正整数解？
I-8. WhenMichael goes up the stairs, he can take 1, 2 or 3 steps at a time. How
many different ways can he go to the 10th step
Michael上台阶时，每步可跨 1级、2级或 3级台阶，他登上第 10级台阶一
共有多少种不同走法？
I-9. Five points are marked on a circle with a circumference of 20cm. The arc length
between any two points is an integer centimeter. The circumference is defined as
the longest arc. What is the maximum number of different arc lengths that can
occur
周长为 20cm的圆周上标有 5个点，任意两点间的弧长均为整数厘米，规定
圆周为最长的弧，那么最多能出现多少种互不相等的弧长？
I-10. How many of the positive divisors of 26! have a units digit of 9
在 26!的所有正因数中，个位为 9的正因数有多少个？
3
Answer（答案）
Question I-1 I-2 I-3 I-4 I-5
题目
Answer 0 337.5 18 16 15
答案
Question I-6 I-7 I-8 I-9 I-10
题目
Answer 6 9 274 19 792
答案
4

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