Difference between revisions of "2023 CMO Problems"
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− | {{CMO box|year=2023|before=[[2022 CMO Problems]]|after=[[2024 CMO Problems]]}} | + | {{CMO box|year=2023|before=[[2022 CMO(CHINA) Problems]]|after=[[2024 CMO(CHINA) Problems]]}} |
Latest revision as of 04:46, 25 May 2024
Contents
[hide]Day 1
Problem 1
Find the smallest real number
Problem 2
Find the largest real number such that for any positive integer and any real numbers , the following inequality holds:
Problem 3
Given a prime number , let . For any , define:
For a non-empty subset of , define:
A subset of is called a "good subset" if and for any subset of with , it holds that .
Find the largest positive integer such that there exist pairwise distinct good subsets of satisfying .
Day 2
Problem 4
Let non-negative real numbers satisfy
Define as the number of elements in the set
Prove that and provide necessary and sufficient conditions for the equality to hold.
Problem 5
In an acute triangle is a point on the extension of . Through , draw lines parallel to and , denoted as and respectively, such that and . Let the circumcircle of intersect at point . Prove: (1) ; (2) .
Problem 6
The numbers are placed on the vertices of a given regular 99 -gon, with each number appearing exactly once. This arrangement is called a "state." Two states are considered "equivalent" if one can be obtained from the other by rotating the 99 -gon in the plane.
Define an "operation" as selecting two adjacent vertices of the 99-gon and swapping the numbers at these vertices. Find the smallest positive integer such that for any two states and , it is possible to transform into a state equivalent to with at most operations.
See Also
2023 CMO(CHINA) (Problems • Resources) | ||
Preceded by 2022 CMO(CHINA) Problems |
Followed by 2024 CMO(CHINA) Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All CMO(CHINA) Problems and Solutions |