I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you :)

Determine all functions with the property that holds for all .

Let be the set of positive integers. Find all functions such that

for all positive integers and .

There is a board with a number written in each cell. Every two neighbour rows sum up to at least , and every two neighbour columns sum up to at most . Find the sum of all numbers on the board.

A square is divided into unit squares. Is it possible to fill each unit square with a number in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?

A quadrilateral with no parallel sides is inscribed in a circle . Circles are inscribed in triangles , respectively. Common external tangents are drawn between and , and , and , and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle . Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.

We denote the number of positive divisors of a positive integer by and the number of distinct prime divisors of by . Let be a positive integer. Prove that there exist infinitely many positive integers such that and does not divide for any positive integers satisfying .

Find all functions such that for any :

Solve the system of equations

Find all functions such that for all