4

Part of 2024 Francophone Mathematical Olympiad

Problems(2)

Find n integers such that a_i divides a_j^2+1 for any i,j

Source: Francophone 2024, Junior P4

Find all integers

n \ge 2

for which there exists

n

a_1,a_2,\dots,a_n \ge 2

such that for all indices

i \ne j

a_i \mid a_j^2+1

number theorynumber theory proposedDivisibilitySequence

Can you pair the divisors of n such that all quotients are close to p?

Source: Francophone 2024, Senior P4

p

be a fixed prime number. Find all integers

n \ge 1

with the following property: One can partition the positive divisors of

n

(d,d')

d<d'

p \mid \left\lfloor \frac{d'}{d}\right\rfloor

number theorynumber theory proposedprime numbersDivisors