Track / Stage 5 / 346 of 400 #946 of 1964

Problem 946

AIME late

Combinatorics Difficulty 5.8 Prove it

32 . $^{+}$ Using first principles, prove that the Hadwiger conjecture holds for $r=4$ .

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

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

None

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

Note: The provided instruction is a meta-instruction and not part of the text to be translated. Since the text to be translated is "None", the translation is also "None". Here is the formatted output as requested:

None

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