Exam III
Strategy
Overview
Since each cast increases mana sources' profit 10 times, our goal is going to be to cast lots of spells and get that multiplier as high as we can, doing some "bursts" along the way to gain additional multipliers from upgrades, sources, etc.
You play this Challenge as an 
Umbramancer, so you tend to make more profits when you are not in Idle Mode than when you are, and you can't earn Compressed Time, Hallowed Clicks, or Void Mana (more on that later).
Summons
First, make decisions about how you want to handle Summons. Since the Spellcraft upgrades for increased summon duration no longer exist, the choice is simple: spend a lot of time waiting for them to cast, or do not cast them. You shouldn't need to cast any of them to complete the challenge.
Pets
Since we want to take advantage of the mechanic that permanently boosts profits based on the "spells cast" statistic, we'll want to use the 
Arcanaworg to charge up spells fast.
You can switch pets after you have charged all the spells you're going to use, but the Arcanaworg's profit multiplier does get pretty good at high levels, and you can get the Arcanaworg's levels very high by using the fact that we have access to so many different spells at once.
In particular:
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If your level is not high enough to equip 
Silence or 
Tempered Arcanomancy, slot 
True Sorcery there instead to raise your character level, and see if that is enough.
Once all these spells have charges, cast them in order from left to right. Since Silence raises the amount of spell shards spent when you consume a charge, your Arcanaworg will level up extremely quickly, to around 130 or higher. You should have no problems gaining spell shards after that, and the profit bonus from the high-leveled Worg might outweigh the benefits of changing pets at this point.
Spells To Use Early
As an Umbramancer, Void Mana is pretty pointless, but we still have access to lots of spells that gain Void Mana or Void Entities. The same is true of Hallowed Clicks. These spells can all be safely spent early on without worry that you might want them for a "burst" later:
Void Lure
Smite
Void Prison
Void Automaton
Summon Winged Night
Relentlessness
Void Syphon
Holy Fervor
Draught of Midas
Void Decompression
Void Radiance
Power of Sacrifice
Battle Trance
Voidforged Armaments
Again, keep in mind some of these are Summons, so you may not want to bother casting them at all to avoid waiting for their long cast time.