There is a contemporary practice of splitting up Sefer T'hilim for on-the-spot recitation of the entire book by a room (real or virtual) full of people. What is the earliest documented source for this practice?
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Recitation of the entire book of Psalms to merit heavenly intervention goes back to Yaakov Avinu, according to Midrash Bereishit Rabba (68). Maharsha (Megilla 11a) writes that this is the source of the practice of Jews to turn to finishing the book of Psalms when beseeching G-d for salvation.
R. Zvi Elimelech Shapira of Dinov (Igra dPirka 348) cites the verse "מי ימלל גבורות ה' ישמיע כל תהלתו" as support for reciting the entire book of Tehillim. The verse's language implies that causing others to say Tehillim is also included (the word yashmia means to cause to be heard). Hence the practice of organizing groups and dividing Tehillim so that the entire book of Tehillim is heard.