2.

Part of 2020 Saint Petersburg Mathematical Olympiad

Problems(3)

sum of some divisors is a square

Source: Saint Petersburg MO 2020 Grade 10 Problem 2

Find all positive integers

n

such that the sum of the squares of the five smallest divisors of

n

Rooks that can not see well

Source: Saint Petersburg MO 2020 Grade 11 Problem 2

A short-sighted rook is a rook that beats all squares in the same column and in the same row for which he can not go more than

60

-steps. What is the maximal amount of short-sighted rooks that don't beat each other that can be put on a

100\times 100

three interesting numbers

Source: Saint Petersburg MO 2020 Grade 9 Problem 2

(a,b,c)

of positive integers we say it is interesting if

c^2+1\mid (a^2+1)(b^2+1)

but none of the

a^2+1, b^2+1

are divisible by

c^2+1

(a,b,c)

be an interesting triple, prove that there are positive integers

u,v

(u,v,c)

is interesting and

uv<c^3