Hi,

After a very disputed game, I reached the endgame with one pawn advantage, but failed to convert it as a win. Can anyone tell me if in this situation White can win?

According to http://www.k4it.de/index.php?topic=egtb&lang=en, yes, you can force mate in 18 moves.

Where did you play this game?  It doesn't look like it was on chess.com.

Anyway, this looks very similar to the lucena position.

One of the key rules of thumb in R+P vs R endings is that, if the defender can get his king in front of the pawn, it's likely to draw. In the example, white controls the promotion square and threatens to queen the pawn if he can check the black king--for example, if the black king is on c2, white queens by 1 Rc8+ and the there's either 1...Rc3 2 Rxc3+ Kxc3 3 b8=Q or 1...Kd2 2 b8=Q Rxb8 3 Rxb8. You'll find this vacating-the-queening square-with-check motif very useful.

In the actual example, you could use this approach as a guide to how to proceed. For example, you could try 1 Kc5. Then 1...Rc3+ 2 Kb4 leaves black without a check and white threatening 3 Rd8+ and promoting. If he tries 2...Rd3 (getting ready to check on d4), 3 Rc8 Rd4+ 4 Kc5 is decisive, since (again) black has no checks to give.

If instead black meets 1 Kc5 with 1...Kc2 (preparing more checks), white wins with 2 Kc6 (guarding the pawn and threatening to slide the Rb8 out of the way) 2...Rc3+ 3 Kd5 Rd3+ 4 Ke4 is decisive, because (again) black has no checks and his king is vulnerable to Rc8+.

R+P v R endings are of extreme importance, and you should study them as much as possible.

thanks alot Mm40, your site is fantastic! I understand what I had to do: bring my King near the rook and the pawn, so that he can protect the pawn and stop check with he rook.  

White can win.

Here isn't it easier just to play Rd8? The pawn is immune- Rxb7 Kc6+ wins the rook.

I saw this exact picture in Seirawan's Winning Chess Tactics this morning. Yes, it is a win, after 1. Rd8 Rxb7 2. Kc6+ picks up the rook.

1.Kc6 Rc3+ 2.Kb6 Rb3+ 3.Ka5 Ra3+ 4.Kb4 and Rd8+ next should promote, doesn't it

but 1...Rd3+ draws

Just to be clear, 1. Rd8 Rd3+ 2. Kc6 Rxd8 3. Kc7 Rh8 4. b8=Q Rxb8 5. Kxb8 and we have bare kings. White has drawn a won position.