2023 CMO(CHINA) Problems/Problem 4
Problem
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.
Solution 1
Let , and . It is easy to see that . Since , there is a solution!
Therefore,
Given that , we have .
Equality is attained when for all and for all . This is perfect. (Note the equality condition of the subsequent inequality. It is achieved when for all and for all , satisfying the subsequent inequality conditions.)
~zata2|szm
See Also
2023 CMO(CHINA) (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All CMO(CHINA) Problems and Solutions |