How to improve your thinking skills?

What methods one could use to prove inequalities IMO problems?

Off the top of your head, you might want to shout out that AM-GM inequality, Cauchy Schwarz inequality. Jensen's inequality are among the "hot" and popular methods that you would consider using to effectively prove the inequality hard IMO problem.

But, do you know we could use the Maclaurin series, integration method, take natural logarithm to both sides of the inequality (just to name a few others) to prove the validity of the inequalities problems as well?

Below is one problem that requires us to think long and hard how to prove it using two of the methods we have brought up to discussion, i.e. to use the Maclaurin formula and also the concept of integration into proving inequality problem.

Prove $\dfrac{\pi}{4}+\dfrac{1}{6}\gt \arctan\left({\dfrac{6}{5}}\right)$.

It's only natural if you immediately think of the inverse tangent formula which is connected with $\pi$, that says:

$\arctan\left({1+x}\right)=\dfrac{\pi}{4}+\dfrac{x}{2}-\dfrac{x^2}{4}+\dfrac{x^3}{12}-\dfrac{x^5}{40}+\dfrac{x^6}{48}-\dfrac{x^7}{112}+\cdots$

Okay, if we replace $x$ by $\dfrac{1}{5}$, what kind of progress or result we will wind up with?

$\arctan\left({1+\dfrac{1}{5}}\right)=\dfrac{\pi}{4}+\dfrac{\dfrac{1}{5}}{2}-\dfrac{\left(\dfrac{1}{5}\right)^2}{4}+\dfrac{\left(\dfrac{1}{5}\right)^3}{12}-\dfrac{\left(\dfrac{1}{5}\right)^5}{40}+\dfrac{\left(\dfrac{1}{5}\right)^6}{48}-\dfrac{\left(\dfrac{1}{5}\right)^7}{112}+\cdots$

$\arctan\left({\dfrac{6}{5}}\right)=\dfrac{\pi}{4}+\dfrac{1}{10}-\dfrac{1}{100}+\dfrac{1}{1500}-\dfrac{1}{125000}+\dfrac{1}{750000}-\dfrac{1}{8750000}+\cdots$

I can see the light at the end of the tunnel how to exploit the above equality to prove the intended inequality. Yeah!

It's about to group the terms on the RHS appropriately, I will leave it to your own imagination and capability to finish the proof, and if you have the doubt and couldn't proceed, I welcome you to post your confusion as the comment and I'll get back to you as soon as I could to guide you through the final step.

Nevertheless, I will show the very important last step for completing the proof by tomorrow. See you all then.