Number theory is one of my favourite topics in Mathematics and in this post you will see a problem involving exponents seeming to be harder has been solved quite easily just using modular arithmetic.
Here is a nice inequality of integrals that I came across a few days ago.
Here is an inequality of triangles
Here is an inequality proved with the help of Cauchy-Schwarz inequality.
Solving a nice high school type integral composed of a function following some condition.