Question 6
Prove that there exist an integer $m\ge2002$ and pairwise distinct positive integers $a_1,a_2,\ldots,a_m$ such that
$$\prod_{i=1}^{m}a_i^2-4\sum_{i=1}^{m}a_i^2$$
is a perfect square.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
$$\prod_{i=1}^{m}a_i^2-4\sum_{i=1}^{m}a_i^2$$
is a perfect square.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.