White to play (45.?) "Medium/Easy"

I almost fell for the trap! I was wondering why this was a two star problem and not easier, and then I found the reason: 45. Qb6+?? -- very tempting because it blocks the check with a check, virtually forcing the trade of queens, leading to a simple won ending for White. Except that it isn't forced! Black has the stalemate resource 45...Ka8! and now if 46. Qxc7 it's stalemate, or 46. Ka6 (only other move) 46...Qc8+! 47. Ka5 Qc7! repeats the position.

Add this to the fact that at first, I thought that 45. b6 was a blunder! I thought that after 45. b6 Qe5+ draws for Black, because it forces 46. Qxe5 fxe5, creating a passed pawn for Black. I was thinking that White couldn't make progress, since advancing the king too far would allow Black's pawn to queen, but a little analysis shows that White queens first (or mates) in all lines:

45. b6! Qe5+ 46. Qxe5 fxe5 47. Kb5 Kb7 (advancing the e-pawn allows White to simply go after it, while White doesn't go after the e-pawn unless it advances, since Black can always play f7-f6) 48. a5 Kb8 (48...f6 49. a6+ is similar) 49. a6 Ka8 50. Kc6 Kb8 (or 50...e4 51. Kc7 e3 52. b7+ Ka7 51. b8=Q+ Kxa6 52. Qb6#) 51. a7+ Ka8 52. Kc7 e4 53. b7+ Kxa7 54. b8=Q+ Ka6 55. Qb6 mate.

So the queen trade after 45...Qe5+ is easy for White. What else? The simple answer is, White wins simply by trading queens at the expense of his b-pawn, advancing the a pawn to a point, then leaving it in favor of gobbling up Black's f-pawns and queening his own f-pawn. For example:

<45.?>

Material: +/♙\ w: ♕ & 4♙ b: ♕ & 3♙* (ENDGAME) *Black has a doubled-pawn.

Candidates: Qb6†... Qb6†??, <[b6]>

-ML-
I think that the main part of this puzzle is to spot that the proziac seemingly winning 45.Qb6† actually draws on the spot. 45....Ka8!! 46.Ka6 (46.Qxc7 1/2) 46....Qc8† 47.Ka5, and now, 47....Qc7! 1/2 of course. That is a VERY nice stalemate and repetition trap. White should remain on the upperhand, however, with:

<45.b6 Qe7> by any means the best reply: Black would want to avoid Qd6† and Qd8†. If he doesn't, for example with 45....Qg3 46.Qd8† Kb6 47.Qc7†, White trades Queens and wins due to the Black King's occupation with the a-pawn.

Today's puzzle of the day is our sixth stalemate in a row. And the game of the day is another stalemate - a repeat of the puzzle of the day of a little over a year ago, which was the third in a sequence of stalemates.

Like buses, policemen and girlfriends, there's never one when you want one, and then three turn up all at once.

I also considered this the best defence, but I think I found a simpler line, thus:

45. b6 Qe7
46. Qc3!