So if we HAVE determined what each character has, what say we get down to the nitty-gritty: who would win in a fight? I say Mustang, his power is highly destructive. Each of his hands serves a different purpose. In the example fight, one hand is in charge of massive explosions, and the other on little snipe shots (and it's not an exaggeration of skill; when focused, he can and has burned out people's eyes.) While Problem Sleuth has very destructive power, most of them require getting rather close to Mustang, which is all but suicide. Really, he only has luck as a total advantage. In fact, despite my hatred of it, I'm going to use that 5-point system I normally use to determine who I believe would win.
Skill: Mustang takes this one. Problem Sleuth is a hard-boiled detective, but his rampage in his self-titled story is the first time he's learning to use these powers. Mustang has spent years learning to control his alchemy; thanks to the efforts of the Hawkeye family, Roy is the only one who can use fire alchemy; the only way to gain experience fighting it is to fight Roy. And Roy very rarely leaves survivors.
Armament: Tied. Roy's fire alchemy is the second-most destructive (flat-out most if he gets a Philosopher's Stone) form of alchemy in the series, losing out only to Kimbley's sociopathic rampages. It serves as both a weapon of mass destruction and as a sniper rifle, capable of various strength outputs and flame sizes. Problem Sleuth's gun is of no use here, but Death's scythe is another matter. Capable of being all sorts of weapons, the scythe gives Sleuth an edge. Or it would, if half the weapons weren't such easy targets to burn away into nothing. Sleuth's only options that won't get flat-out destroyed are probably guns, which aren't of much use; the atomic bomb, which would kill them both; and melee weapons. Those are his best bet, but even then he has to be very careful not to be incinerated.
Endurance: Roy wins out here, but not in the conventional sense. In vampire form, Sleuth can probably take plenty more damage. The difference here is that Death's Scythe would not put out as much damage as Roy's fire. While Sleuth could probably take one good hit from the flames before kicking the bucket, Roy has the advantage of being up against a weapon that more or less sticks to conventional terms. If he's impaled, he can still keep fighting, as he has proven in the past. If he is bleeding out, he can fix himself up, though at that point he's on a timer to end the match quickly. It's not by much, and it's not in the way you think, but Roy can endure more of what Problem Sleuth dishes out than vice-versa.
Agility: Problem Sleuth. While Roy is as fast as an athletic human or well-trained soldier, he's still just that: human. Problem Sleuth, for this battle, is in vampire form and therefore much quicker than Roy. Instead, Roy must rely on keeping Problem Sleuth at bay with massive firewalls, which will not be easy due to the sheer speed and power of the vampiric form.
Preparation: This one comes down more to psychology than anything else. Roy is a rather silly man at his worst, but when push comes to shove you can tell there's always a hidden goal behind his seemingly stupid actions, something to further his own goals. With no way to Obfuscate Stupidity around here, Roy will be forced to get serious, spending all his time studying or training. Since he's already in more or less peak condition, time will be spent learning: all of his foe's moves, and deciding how best to counter each. Problem Sleuth would probably take up at least six hours of study, being a wild card in this tournament. He would be fully prepared.
Problem Sleuth, only the other hand, is a sillier sort than Roy, even. While he tends to further his own goals by doing them, most of his actions are rather nonsensical and arbitrary. Based on the time spent in the Problem Sleuth adventure, we can be more or less guaranteed he'd spend a day and a half just goofing around in his office. Roy would probably get a few hours of study though, as his powers are rather specifically tailored to kick the Sleuth's ass. As far as actual preparation goes, Roy Mustang would take the matter more seriously.
With a score of 3.5/1.5, Roy Mustang beats Problem Sleuth in the scoring aspect. I have already presented my case as to how the actual fight would go down, so that's it from me. I hope the judge will see the obvious choice for victory in this fight.