You need to control the sources that dealt damage only at the times when they dealt damage. If they die or change control after that point, the "to solve" ability will still trigger at the beginning of your end step as long as you still control Case of the Burning Masks.

Activated and triggered abilities are not themselves sources of damage. The source of an activated ability is the object whose ability was activated, and the source of a triggered ability (other than a delayed triggered ability) is the object whose ability triggered.

If multiple creatures you control deal combat damage, each one counts as a source that dealt damage that turn. A single creature that deals combat damage multiple times in a turn (due to double strike or additional combats) still counts as only one source.

Something that changes zones becomes a new object in that zone and will therefore be a new source. For example, if you cast Flametongue Kavu, deal damage with its enters-the-battlefield ability, exile and return it to the battlefield with Momentary Blink, deal damage again, then cast Momentary Blink with flashback on Flametongue Kavu to repeat the process one more time, you've fulfilled Case of the Burning Masks's "to solve" trigger condition. 

A permanent that deals damage multiple times without changing zones still counts as only one source. For example, if you attack with Incinerator of the Guilty and deal combat damage with it, use its triggered ability to deal damage, and then use Hard-Hitting Question to have it deal damage to another creature, it's still just one source even though it dealt damage three times during the turn.

You pay all costs and follow all normal timing rules for the card played from exile with Case of the Burning Masks's last ability. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.

Each Case has two special keyword abilities: to solve and solved.

"To Solve — [condition]" means "At the beginning of your end step, if [condition] and this Case is not solved, it becomes solved."

The meaning of "solved" differs based on what type of ability follows it. "Solved — [activated ability]" means "[Activated ability]. Activate only if this Case is solved." Activated abilities contain a colon. They're generally written "[Cost]: [Effect]."

"Solved — [Triggered ability]" means "[Triggered ability]. This ability triggers only if this Case is solved." Triggered abilities use the word "when," "whenever," or "at." They're often written as "[Trigger condition], [effect]."

"Solved — [static ability]" means "As long as this Case is solved, [static ability]." Static abilities are written as statements, such as "Creatures you control get +1/+1" or "Instant and sorcery spells you cast cost {1} less to cast."

"To solve" abilities will check for their condition twice: once when the ability would trigger, and once when it resolves. If the condition isn't true at the beginning of your end step, the ability won't trigger at all. If the condition isn't true when the ability resolves, the Case won't become solved.

Once a Case becomes solved, it stays solved until it leaves the battlefield.

Cases don't lose their other abilities when they become solved.

Being solved is not part of a permanent's copiable values. A permanent that becomes a copy of a solved Case is not solved. A solved Case that somehow becomes a copy of a different Case stays solved.