Track / Stage 5 / 117 of 400 #717 of 1964

Problem 717

AIME late

Algebra Difficulty 5.3 Find the answer

7. (1) Find the number of positive integer solutions $(m, n, r)$ to the indeterminate equation $m n + n r + m r = 2(m + n + r)$ ;
(2) For a given integer $k (k > 1)$ , prove that the indeterminate equation $m n + n r + m r = k(m + n + r)$ has at least $3 k + 1$ sets of positive integer solutions $(m, n, r)$ .
(Wu Weizhao)

The source for this one didn't record the answer, so there is nothing to check what you type against. Work it on paper and mark yourself against the solution below.

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.