What is the source for pushing away the convert three times?
I'm also interested in the exact quote.
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It's in the Midrash [Ruth Rabbah 2:16]:
Rabbi Shmuel bar Nahmani said in the name of Rav Yehudah ben Hanina: Three times is it written [in the Book of Ruth] "turn back", corresponding to the three times that a would-be convert is repulsed;
וַתֹּ֤אמֶר נָֽעֳמִי֙ לִשְׁתֵּ֣י כַלֹּתֶ֔יהָ לֵ֣כְנָה שֹּׁ֔בְנָה אִשָּׁ֖ה לְבֵ֣ית אִמָּ֑הּ יַ֣עַשׂ (כתיב יַ֣עַשׂה) יְהֹוָ֤ה עִמָּכֶם֙ חֶ֔סֶד כַּֽאֲשֶׁ֧ר עֲשִׂיתֶ֛ם עִם־הַמֵּתִ֖ים וְעִמָּדִֽי:
And Naomi said to [Ruth and Orpah], Go, return each of you to her mother's house
וַתֹּ֤אמֶר נָֽעֳמִי֙ שֹׁ֣בְנָה בְנֹתַ֔י לָ֥מָּה תֵלַ֖כְנָה עִמִּ֑י הַֽעוֹד־לִ֤י בָנִים֙ בְּֽמֵעַ֔י וְהָי֥וּ לָכֶ֖ם לַֽאֲנָשִֽׁים:
And Naomi said, Turn back, my daughters; why will you go with me?
שֹׁ֤בְנָה בְנֹתַי֙ לֵ֔כְן ָ כִּ֥י זָקַ֖נְתִּי מִֽהְי֣וֹת לְאִ֑ישׁ כִּ֤י אָמַ֨רְתִּי֙ יֶשׁ־לִ֣י תִקְוָ֔ה גַּ֣ם הָיִ֤יתִי הַלַּ֨יְלָה֙ לְאִ֔ישׁ וְגַ֖ם יָלַ֥דְתִּי בָנִֽים:
Turn back, my daughters, go your way.
but if he persists after that, he is accepted.
וַתֵּ֔רֶא כִּֽי־מִתְאַמֶּ֥צֶת הִ֖יא לָלֶ֣כֶת אִתָּ֑הּ וַתֶּחְדַּ֖ל לְדַבֵּ֥ר אֵלֶֽיהָ:
When [Naomi] saw that [Ruth] was determined to go with her, she stopped [objecting].
-R. Isaac said: …A man should rebuff with his left hand, but bring near with the right.