Here is a solution that does not rely (too much) on softwares. I will be using the known values of the sums $\small{\displaystyle \sum^\infty_{n=1}\frac{H_n}{n2^n},\ \sum^\infty_{n=1}\frac{H_n}{n^22^n},\ \sum^\infty_{n=1}\frac{H_n}{n^32^n}}$. Let $$\mathcal{S}=\sum^\infty_{n=1}\frac{H_n}{n^42^n}...

This question came in RMO, an olympiad in India. I solved it but with the assumption that the lines are parallel, though we are not given this info in the question. Let ABC be a triangle. AD be a perpendicular from A to BC. Let there be 3 other lines intersecting AD at K,L,M such that AK = KL = ...

I have tried many methods but do not know how to integrate this: $$ \int \frac{1}{x\ln x + 7\ln x} dx $$ with respect to x.