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58 Qf6+ gives perpetual check

#4 No that is right.
56...Kd4 was the only move that allowed the perpetual, all other king moves would have won.

8/8/8/8/8/5K2/1q1k4/q5Q1 w - - 0 1 at this position Qg5+ draws he is in time trouble he miss Qf6+

#6 that is correct. Qa1 + Qb2 are not much stronger than a single queen, Ba1 is almost like as if it were a bishop. Perpetual checks are a major saving resource in queen endings.
That is also why on move 56 the king had to flee on a light square, so that the queens could help defend against the perpetual along the dark square diagonal a1-h8.

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