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Problem 284

AMC 12 late, AIME early

Combinatorics Difficulty 4.5 Find the answer

4. Four volleyball teams participate in a round-robin tournament, meaning each pair of teams plays one match. If the score of a match is $3: 0$ or $3: 1$ , the winning team gets 3 points, and the losing team gets 0 points; if the score is $3: 2$ , the winning team gets 2 points, and the losing team gets 1 point. If after the matches, the points of each team are exactly four consecutive natural numbers, then the points of the first place team is $(\quad)$ points.
(A) 3
(B) 4
(C) 5
(D) 6

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.