The answer of the fifteenth riddle - Jealousy's Fault
The first paragraph.
The legend of the man and the snow
The fake illness which never truly what it seemed
And a body double with almost identical face showed
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» There's this legend mentioned in the episode, it also indicated the place Conan and Co were, in the place which had snow. Then someone pretended to be ill, so he/she would be able to disguised into another person that had the similar appearance.
The second paragraph.
Played the role of two person
Pretended all the whole situation
The seemingly suicide would never was a suicide
And everything would all reconcile
With no one's happiness
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» That someone acted like he/she was the another person and fooled people that it was really who they thought that someone was. Since this is DC, we can suspect that someone killed the another and made it looked like a suicide (again).
The third paragraph.
Jealously was the one took over
Convinced person's insanity came to play
Took away the one who beloved
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» And this's probably the motive: the jealously.
The Conclusion
The very first line is the crucial hint, referring to the Yuki Onna legend mentioned in the episode 94 (Hey, hshh-san! Yes, you're right!).
The body double referred to Kinoshita Akiko-san and Asanuma Yoko-san, both an actress. Yoko-san killed Akiko-san by buried her in the snow, caused her to froze to death. Yoko-san then tricked everyone to believed that Akiko-san was in her room all the time because of her (forge) illness, by playing the role of both persons and the fake letter. The motive is jealousy, because Akiko-san always got the fame on the job and Yoko-san is always "the second".
Go Go GO! XD