2023 CMO(CHINA) Problems/Problem 2
Problem
Find the largest real number such that for any positive integer
and any real numbers
, the following inequality holds:
Solution 1
Define the matrix
as follows:
The problem simplifies to finding the smallest real number such that for all
and any vector
, the following inequality holds:
In other words, find the real number such that:
Given that
is not easily invertible directly, but is invertible (as it is a sparse matrix):
Since the inverse has non-zero entries:
For the eigenvalues of
:
Thus:
Given that
is invertible,
, therefore:
For a specific , we have:
In conclusion:
Therefore, the maximum value of the real number is
.
~dalian xes|szm
See also
2023 CMO(CHINA) (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All CMO(CHINA) Problems and Solutions |