Do you have the correct position in front? 
39.Bd2 Kd6 40.b4 Ke5 is not legitimate because of the White rook on h5.
Moreover you also have to find a Black win after 39.Ke3 Kd6 40.Bc3 and 41.b4.
My conclusion remains: a solid draw.

"This has really been interesting"
What do you mean? It still is. I will take a look at 39.Bd2 Kd6 40.b4 Ke5.
I do not agree with you on the subject of trading pieces. My point is, that after exchanging rooks White gets chances to create a second passed pawn on the a-file and can advance the passed f-pawn. The king will block on d3. This way White makes optimal use of the long-ranged bishop. I think you underestimate the power of the bishop vs the knight in endings with pawns on all fronts.
At the other hand I must admit, that I feel more comfortable in light-piece endings than in rook endings, so I am not sure who is prejudiced after all.
I must say, that I am a bit disappointed about the lack of interest among other regular writers. First I read so many posts about the importance of endgame skills. Now I am one of the very few involved in and endgame debate?!

What about 39.Bd2 Kd6 40.b4 b6 41.Rh6 Ke5 42.Rh5+ Kd4 43.Rh6 as Re7 44.Be3+ Kc4 45.Bxb6 Kxb4 46.Rh4+ is clearly better for White?
And 39.Ke3 Kd6 40.Bc3 Rf7 41.b4 is not solved either - here no march via d4.

Kevinludwig: I am interested indeed - I guess, that White starts with 39.f3 ?

Smyslov_fan: I am looking forward to it. It does not happen that often, that I get endgame training for free!

I have another ending for everyone to consider:
White has pawns on f2, g3, h4, a6. Ke3, Ra7.
Black has pawns on f7, g6, h5, Kf6, Ra2.
The position occurred in Tal-Gipslis, 1983. Tal played 1. Ra8 and the game was soon drawn. But it turns out that white is winning. 

There is actually published analysis on this, and I first saw it a few years ago, and spent a while looking at it. The position is interesting, because it is the sort of thing that might come up in a real game. I'll tell where the analysis is online if there is any interest.

For one thing White must prevent the pawn roller and play b4. Without a board I am not certain, but 39.Bd2 Kd6 40.b4 is the first thing I will look at. Compare 39.Ke3 Kd6 40.Bc3 Rf7 41.b4. I am not sure, if the bishop is better on c3 or on d2.
Fritz' 39.Bd2 Kd6 40.a5? definitely sucks, as it rules out any counterplay with b4-b5 and puts a pawn on a black square. So does b2-b4, but the difference is that c6-c5 is hindered and that b4-b5 remains possible.

A few hours later, with a board, it is clear to me that b2-b4 preventing c6-c5 is the key move to equality.

Thanks, MNb, for including the entire game.  BTW, it was the 8th and penultimate, not final round.   

I made an even more important mistake:  I meant to type:  e6, not e7.  With the pawn on e7, it is 

BLACK to MOVE

MNb, your suggested improvement certainly seems to gain a tempo on the line played, and Fritz prefers it to 39.Bb4+  However, I don't see how it actually improves White's position enough to save the game.  Black is threatening to attack the f-file with the N and R.  In this particular game, I would definitely prefer the centralised N to the peripheral B, and Korchnoi obviously felt the same way. 

What this means is that I think if White has salvation it will be precisely in the Rook and Pawn ending that you found so distasteful.  White neeeds to slow down the c-pawn.  I think his fatal error was playing b4, allowing Black to create the monster passers.

My main question is, after 34. e7 Ke7 35.Rxf5+ Kxe7 36.Rh5 Rg8+ 37.Kf2 Rg7 38.Bxa5 Ne6 (all in the game)

What does White do after 39.Bd2! (MNb) Kd6, as Ponomariov would almost certainly have continued?  How does he avoid getting steamrolled in the center and Q-side with the play 40...b5, 41...c5 42....c4?

Does White try to play passively with b3?  I don't see that working out very well either.  Ideas?

Wow, that was easy!

I just went to the Geller Memorial Tournament games that just finished, and came up with some very interesting examples.  In rounds four and five respectively, Viktor Korchnoi lost, then won a Q+B+Ps ending with same colored Bishops and minus one pawn.  Instead of those two games, I decided on a third Korchnoi game, his last round loss to Ruslan Ponomariov.  I took a *very* quick look at the game in Fritz, and decided that it's sufficiently complex and interesting enough for us to analyse and still be instructional.

(If someone could provide a diagram, I'd appreciate it.)

Korchnoi-Ponomariov, 2005
White:  Kg1 Re5 Bd2
            Ps:  a4 b2 e7 f3 h2
Black:  Kg8 Re8 Nc7
            Ps:  a5 b7 c6 d5 f5 h7

White to move.

Intro info:  White lost this game after temporarily losing  a pawn, and then trading into a pure rook ending to regain the pawn.  However, Black had connected passers on the c+d files, which ended up deciding the game.  Let's see if we can't find any improvements for GM Korchnoi!

I for one find endgames pretty boring & dry stuff.

But though I hate to say it..... I hv to agree with Capa. "Study the endgame 1st" The endings are an essential part of the game these days cos defensive techniques and middlegame play are so good there are no longer anymore rapid kills in the middlegame as seen in the days of Morphy et al. (maybe even Tal)

I'm about 1700 rating. When I was competitive like 20 yrs ago I payed a lot of attention to openings only. My regret was that i did not study middlegame and endings. So i lost quite a nomber of games. Very often also i would lose even though i had a better position cos i didnt know how to handle the ending.

So if you are ambitious and are shooting for the GM & Im titles, there is really no excuse to avoid studying it. or if you are like me... just intending to play a good game and beat friends , internet players or even the computer, it is still essential if you want to win. Many an internet game i won even though i blundered badly in the opening or middlegame cos my opponent couldnt handle the ending properly (not that i was any good, its just that i knew a minute something & he knew 0). But all i know abt the ending is still just scratching the suface or less. I still dont quite get Kotov's statement in 'Think like a grandmaster' when he speaks of the ending as thinking in terms of schemes (or something to that effect)

perhaps its a good idea to have a forum on endings (or even middlegames) if possible. I personally feel that endings and middlegame are actually more impt than the opening, cos proper play there would ensure a win. The opening & all its latest lines is just a means to get to the middlegame in a so called favourable position. But how one handles that so called advantage to convert it to a win is quite another story.  In this sense I pretty much agree with Klick.& Akavall