6

Part of 2021 Tuymaada Olympiad

Problems(2)

Bound of prime factors implying product of many factors

Source: Tuymaada 2021/J6

y>1

and positive integer

n \leq y^{50}

such that all prime divisors of

n

y

n

is a product of

99

positive integer factors (not necessarily primes) not exceeding

y

Rearrange the rows and columns to move marked squares above the main diagonal

Source: Tuymaada 2021 Senior P6

n\times n

n>1

k

unit squares are marked.One wants to rearrange rows and columns so that all the marked unit squares are above the main diagonal or on it.For what maximum

k

is it always possible?