1

Part of 2023 New Zealand MO

Problems(2)

2023 employees in the office

Source: 2023 NZMO - New Zealand Maths Olympiad Round 1 p1

There are 2023 employees in the office, each of them knowing exactly

1686

of the others. For any pair of employees they either both know each other or both don’t know each other. Prove that we can find

7

employees each of them knowing all

6

p! x q! x r! is perfect square if p > q > r > 1

Source: 2023 NZMO - New Zealand Maths Olympiad Round 2 p1

For any positive integer

n

n! = 1\times 2\times 3\times ... \times n

. Do there exist infinitely many triples

(p, q, r)

, of positive integers with

p > q > r > 1

such that the product

p! \cdot q! \cdot r!

$ is a perfect square?

number theoryPerfect Squares