I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here.

In the Wikipedia page on Ramanujan, there is a link to a collection of problems posed by him. The page has a collection of about sixty problems which have appeared in the Journal of the Indian Mathematical Society.

While I was browsing through them at random, I came across this, which I recognized as the Brocard-Ramanujan problem. So,

1. Are all these problems solved? Or are there more unsolved problems among them?
2. Why were the problems posed exactly? Was there some sort of contest in the journal back then?

Here is one problem from the list. (I just cannot resist posting it here!)

Let $AB$ be a diameter and $BC$ be a chord of a circle $ABC$. Bisect the minor arc $BC$ at $M$; and draw a chord $BN$ equal to half the length of the chord $BC$. Join $AM$. Describe two circles with $A$ and $B$ as centers and $AM$ and $BN$ as radii cutting each other at $S$ and $S'$ and cutting the given circle again at the points $M'$ and $N'$ respectively. Join $AN$ and $BM$ intersecting at $R$ and also join $AN'$ and $BM'$ intersecting at $R'$. Through $B$, draw a tangent to the given circle meeting $AM$ and $AM'$ produced at $Q$ and $Q'$ respectively. Produce $AN$ and $M'B$ to meet at $P$ and also produce $AN'$ and $MB$ to meet at $P'$. Show that the eight points $PQRSS'R'Q'P'$ are cyclic and that the circle passing through these points is orthogonal to the given circle $ABC$.

How does one even begin to prove this? I've tried to construct the entire thing using compass and straightedge, but the resulting mess only added to my confusion. Here is the link to the original problem.

Thanks a lot in advance!

EDIT: This problem is mentioned in the link provided in Andrey Rekalo's answer below in pp. 34-35. The solution is given in the second link in pp. 244-246.