Competition problem — Number theory

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Number theory Difficulty 5.5 AIME, harder Prove it Mongolia

Find all positive integers $n$ such that there exists a positive integer with $n^2$ divisors (including $1$ and itself) and an arrangement of those divisors (each divisor is used once) in an $n \times n$ grid, where the sum of the divisors in each row and column is the same.

(Battsengel B., Bayarmagnai G.)

Solution

*Answer: $n=1$ .*

Suppose there exists $n \ge 2$ satisfying both requirements. Let $a_j$ be the largest number in the $j$ -th column for $1 \le j \le n$ . Without loss of generality, we may assume that $a_1 > a_2 > \dots > a_n$ . Thus $j a_1 \le a_j$ since $a_j$ divides $a_1$ . Hence it is clear that the sum of integers in the $n$ -th column is not greater than
$\sum_{i=1}^{n} \left( \frac{a_i}{n} - i \right) = a_1 - \frac{n(n+1)}{2}.$

Therefore the sum of integers in the first column is greater than that of the $n$ -th column.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.