white to play and win
Wow, it took me a long time to find out what happens after 2...Rxa3 (looks like I'm a bit tired hehe). It's amazing! I definitely have to remember this position :)
As pfren suggests the black rooks should be placed on g2 (or h2).
There is an important variation for white. After 1. Qa3+ Ra2 he can try 2. Qc5. Black would fail then with 2. .. b2+? 3. Kd2+ b1Q+ 4. Ke1 and white wins. But black can repeat moves with 2. .. Rh2! instead forcing white to find the main line.
Wow, it took me a long time to find out what happens after 2...Rxa3 (looks like I'm a bit tired hehe). It's amazing! I definitely have to remember this position :)
I found your solution, but I have doubts about 1. Rf1 and 1. Rd4 Won't these moves win as well?
That accounts for three of White's five winning moves. Without changing the position, it would have worked as a mate in 13 problem, but the OP's solution would have been wrong.
Of course, relocating the rook and keeping it as a study is much better.
I have a question. If the Black rook is placed on h2, before the key move Qa3+ White can first force a repetition with Rd2. Does that count as a second key move? It doesn't seem to me that it should.
It counts as a minute minus if at all. Lots of compositions have phases where white can make a 'circling' detour before resuming the proper course. It is one of the inevitabilities of study composition and does not count as a serious dual.
I knew you would come up with this question one of these days :D. There are a number of related duals but this one is certainly the most harmless. The option for white to play 2. Qc5 is exactly such a detour as 1. Rd2.
Note that these duals also have a positive side. They may be viewed as 'failed attempts' and therefore have the feel of a 'white try'. If the chess rules were a little more strict on repetitions - by drawing the 1st repeat - then these moves would not exist as duals at all but only as valuable tries!
I wasn't concerned about the second spurious dual you mentioned, but I wondered if one occurring on the key move might be looked at differently.