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）
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. Simplify: 3 2026 678 675 3 2026 678 675 .
化简： 3 2026 678 675 3 2026 678 675 。
I-2. Given that the positive integer n satisfies n!=n3 – n, find the value of n.
已知正整数 n满足 n!=n3 – n，求 n的值。
I-3. Let A be a non-empty set whose elements are all positive integers less than 6. The
set A has the following property: if a∈A, then |6 – 2a|∈A. Find the number of
possible sets A that satisfy this condition.
已知非空集合 A中的元素均为小于 6的正整数，且具有性质：若 a∈A，则
|6 – 2a|∈A。求满足条件的集合 A的个数。
1
I-4. Let P(x) = x3+bx2+cx+d, where P(1)=2026 and P(2)=4052. Find the value of P(5)
– P(–2).
设 P(x) = x3+bx2+cx+d且 P(1) = 2026，P(2) = 4052，求 P(5) – P(–2)的值。
I-5. How many ways are there to place 8 identical balls into 3 distinct boxes, allowing
some boxes to be empty
把 8个相同的球放入 3个不同的盒子中，允许有空盒的放法有多少种？
I-6. Positive integers x and y satisfy x x 1 16y2 38y 35 , find the value of x + y.
正整数 x，y满足 x x 1 16y2 38y 35，求 x+y的值。
I-7. As shown in the figure, ABCD is a convex quadrilateral. M and N are the
midpoints of AC and BD, respectively, and P is the midpoint of AB. Given that
AD=10, BC=16, ∠MPN=60°, and AC2+BD2=400, find the value of AB2+CD2.
如图，凸四边形 ABCD中，M，N分别为 AC，BD的中点，P为 AB的中点。
若 AD=10，BC=16，∠MPN=60°，AC2+BD2=400，求 AB2+CD2的值。
3
I-8. The general term of the sequence {a } is a n 1n n 3 . Find the value of a2a3…a10.n 1
n3 1
已知数列{an}的通项公式为 an 3 ，求 a2a3…a10的值。n 1
2
I-9. Among the natural numbers from 1 to 50, choose m numbers (m>1). Find the
minimum value of m such that among these m numbers, there must be at least one
number that divides the product of the remaining m 1 numbers.
从自然数 1至 50中任意选出 m(m＞1)个数，在选出的这 m个数中必有 1个
数可以整除其他 m – 1个数的乘积，求 m的最小值。
I-10. A frustum of a right circular cone has upper and lower base areas of 9π and
25π, respectively, and a height of 6. A plane parallel to the two bases cuts the
frustum into two smaller frustums of equal volume. Find the radius of the circular
cross section.
圆台的上、下底面的面积分别为 9π和 25π，高为 6。用一个平行于上、下底
面的平面截这个圆台，所得的两个圆台的体积相等。求截面圆的半径。
3
Answer（答案）
Question
I-1 I-2 I-3 I-4 I-5 I-6 I-7 I-8 I-9 I-10
题目
Answer 37
2 5 5 14266 45 10 240 16 3 76
答案 55
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