So I was reading the solution of the following problem.
A $100X100$ array is filled with numbers from ${1,2,...,100}$ ,such that each number appears exactly 100 times.Prove that there is some row or column which contains atleast 10 different numbers.
In the solution it says that we can prove through AM-GM that each number appears in at least 20 rows-columns.I have proved this another way but i can't find how to apply AM-GM. Any help will be appreciated.
Let us say that the number appears in $a$ rows and $b$ columns. That means that it appears at most $ab$ times (intersections between these allowed rows and columns). By AM-GM, you have:
$$100 \le ab \le \left(\frac{a+b}2\right)^2.$$
Therefore, $10 \le (a+b)/2$ which gives $a+b \ge 20$ as desired.