I think that Iranian mathematicians have lots ot things to show. There are many problems in Iranian Math Olympiad taken from European countries but math unions people. I'm sorry for offtopic but I think that if you put as more problems in Resources section in this site from Iranian Math Olympiad will be very good. I like this competition. Thank you again!

borislav_mirchev wrote: I think that Iranian mathematicians have lots ot things to show. There are many problems in Iranian Math Olympiad taken from European countries but math unions people. I'm sorry for offtopic but I think that if you put as more problems in Resources section in this site from Iranian Math Olympiad will be very good. I like this competition. Thank you again! It is not so. All of the problems are designed in large sessions and after long sessions. They are not copied from European countries problem. If it is, it's is accidental. P.S. I believe the same as you : Math unite people together.

I don't know what is the situation now, but I took in mind problems in Mohammad Katoozian web site. [1997-1999] - it is maybe first round. http://www.geocities.com/CollegePark/Lounge/5284/index.html 1998-1-i saw this in Argentinian math olympiad. 1999-1 - seems to be from a russian olympiad, 5 is a theorem, and maybe 2, 4 - saw something like this in Irish math olympiad, problem 6 is from APMO or Hungary Israel math competition i think, I saw this in mathlinks.

Those problems are the old ones. But I tell you after 2001 or 2002 nothing is copied from other Olympiads.

It is very good! I think it also take lots of time and lots of people are engaged with the competition. I think in future Iran will have very strong team, traditions and success. As I understood you are engaged in the olympiad. I wish success to you and your team!

(a) Let $ R_{1},R_{2}$ be two intertwining rings, and let $ C_{1},C_{2}$ be the outer boundary circles of $ R_{1},R_{2}$ respectively. Finally, let $ \pi$ be the plane in which $ R_{1}$ lies. $ C_{2}$ cuts the portion of $ \pi$ enclosed by the inner boundary circle of $ R_{1}$ in some point $ P$, whose distance from $ C_{1}$ must thus be at least $ 1$. It follows that the Hausdorff distance between $ C_{1}$ and $ C_{2}$ is at least $ 1$. This holds for any two rings $ R_{1},R_{2}$ in our set, so the outer circles of such a family form a discrete set in the metric space of all compact subsets of $ \mathbb R^{3}$ with the Hausdorff metric. Since this metric space is separable, we're done: any set of mutually intertwining rings with width $ 1$ is at most countable. (b) The Hopf map provides an example. It's a map $ p:\mathbb S^{3}\to\mathbb S^{2}$ such that the preimage of any point in $ \mathbb S^{2}$ is a circle, and such that any two distinct preimages are linked. There are pictures on that website of mutually linked circles on a torus obtained by restricting this map.

Official Solution: The answer for (a) is no. Assume that two rings $ A,A'$ with centers $ O,O'$ are tied together. We will show that $ |OO'|\geq 1$. Let $ R,R'$ denote the inner radii of the rings. Assume that $ R'\geq R$. The plane of the ring $ A'$ intersects the ring $ A$, in two intervals, one inside the ring, and the other outside. Let $ U$ be a point on the outer border of $ A'$ that is on the plane and inside the ring $ A$. Then $ R'+1=|O'U|\leq |O'O|+|OU|\leq |OO'|+R\leq |OO'|+R'$ which shows that $ |OO'|\geq 1$. Now for each ring with $ O$ as the center, consider the sphere with radius $ \frac 13$ around $ O$. This sphere contains a point with rational coordinates. So there is an injective mapping from the rings to the rational points. (It is injective because no two such spheres intersect) So the number of rings must be countable. The answer for (b) is yes. Consider a circle with center $ (0,0,0)$ lying in the xy plane and having radius $ 1$. For each $ 0\leq\theta\leq0.1$, move the circle by $ \theta$ along the positive direction of the x axis, and rotate it along this axis by $ \theta$ radians. It is obvious that the rings constructed in this way are pairwise tied to each other. They are in one-to-one correspondence to $ [0,0.1]$ which is uncountable.