2023 IMO Problems/Problem 4
Problem
Let be pairwise different positive real numbers such that
is an integer for every
. Prove that
.
Video Solution
https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems]
Solution
We solve for in terms of
and
Again, by AM-GM, the above equation becomes
Hence, but equality is achieved only when
and
are equal. They can never be equal because there are no two equal
So