Well, the first thing I thought of was for the white to play Rf3, to be followed with Qf7+ and moderate grief to be inflicted on the black thereafter, but nothing flashy.

THEN it dawned on me that the puzzle calls for the black to move first :-)

But, as a result of the initial misunderstanding, 31...Rxa2 did not enter the considerations at all, since the same moves as above are a pretty good counterplay. I started with he idea of Rxf2, noticed the same stuff as <dzechiel> and chased the white Q away from defending f1, finding the same line as played in the game up to move 33, which seemed good enough.

31...♖d1 then if 32 ♖xd1 ♖xd1+ now the ♕ must prevent mate with 33 ♕f1 and now the ♕ is exchaned for the ♖ 33...♖xf1+ 34 ♔xf1 and Black retains the ♕ against the remaining White ♖

But White can also play 32 ♖c1 which prevents the back rank threat and returns to equality. So that's not it.

31...♖xf2 seems to suffer a similar fate and {31...Qxf2 is even worse.

Ahah! - I think I see it.

31...♖8d4 and now the ♕ has only one safe square to escape to.

32 ♕e6 allows 32...♖d1 and now 33 ♖c1 is met with 33...♕xc1 34 ♖xd1 ♖xd1+ 35 ♔h2 ♕f4+ 36 g3 ♕xf2#

Time to check.

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I think my line still works after the diversion 33 ♕xe7+ ♔h6 which doesn't accomplish anything.

Time to check.

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The whole combination must center around the theme ♖f2:.

The problem is that playing ♖f2: right away is bad:

31. ... ♖f2:? 32. ♖f2: ♖d1+ 33. ♖f1!

The hindrance is the queen on c4 which protects f1 and therefore makes 33. ♖f1 possible.

It will be necessary to drive away the queen with tempo. Only 31. ... ♖8d4 fulfils those criteria since the d2-rook is needed for the attack against f2.

So it has to be

31. ... ♖8d4

32. ♕e6 (Only move, 32. ♖f3? ef , the pawn is not pinned anymore)

Now it is interesting:

Can we really play 32. ... ♖f2: or will there be a perpetual (or even a win after forks) for White after 33. ♕e7:+?

Let's see:

32. ... ♖f2: 33. ♕e7:+ ♔h6 and there is no perpetual:

34. ♕f8+? ♕f8: 35. ♖f2: ♖d1+ 36. ♔h2 ♕f2:

34. ♕h4+? ♕h4: 35. ♖f2: ♖d1+ 36. ♔e2 ♖d2+! 37. ♔d2: ♕f2:+

Black can not take on f2, either:

34. ♖f2:? ♖d1+ 35. ♖f1 (Only move) ♖f1:#

Second Part:

Obviously there is some more to it than that:

White can let e7 untouched, decline the sacrifice (♖cc1 or ♖f3 come to mind) and wait for a convenient moment to take on e7 or he can play ♕e7:+ right away and decline only after ♔h6, which might make a difference in some lines.

Until now we have:

31. ... ♖8d4 32. ♕e6 ♖f2:

I. The first branch: ♕e7:+ right away

33. ♕e7:+ ♔h6

1. 34. ♖cc1? Δ to protect f1 is refuted by 34. ... ♖g2:+! 35. ♔g2: (♔h1 ♕h2#) ♖d2+ and either

36. ♔g1(h1) ♕h2# or 36. ♖f2 ♕f2: + 37. ♔h1 ♕h2#

2. Any move of the f1-rook along the first rank (Δ to keep f1 protected this way) is again refuted by 34. ... ♖g2:+ with mate according to I.1.

3. The more interesting 34. ♖f3? (Δ 36. ... ♖f1:+) is refuted by 34. ... ef 35. ♖f2: (What else?) ♖d1+ 36. ♖f1 (Only move) f2+! 37. ♔h1 ♖f1:#

II. The second branch: Leaving the queen on e6

After our analysis of I. we see that this will make no difference since the queen on e6 instead of e7 does not help in any line.

Conclusion:

If nothing elementary was overlooked in the above analysis an exemplary main line with the most pitfalls for Black could be: