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 |