4

Part of 2024 Saint Petersburg Mathematical Olympiad

Problems(3)

caa...ab is always composite

Source: Saint Petersburg olympiad 2024, 11.4

101

a

and an arbitrary positive integer

b

. Prove that there is at most a

102

-digit positive integer

c

such that any number of the form

\overline{caaa \dots ab}

At most N^2 pairs has common root

Source: Saint Petersburg olympiad 2024, 10.4

Let's consider all possible quadratic trinomials of the form

x^2 + ax + b

a

b

are positive integers not exceeding some positive integer

N

. Prove that the number of pairs of such trinomials having a common root does not exceed

N^2

Volleyballers throws balls

Source: Saint Petersburg olympiad 2024, 9.4

The coach lined up

200

volleyball players and gave them

m

balls (each volleyball player could get any number of balls). From time to time, one of the volleyball players throws the ball to another (and he catches it). After a while, it turned out that of any two volleyball players, the left one threw the ball to the right exactly twice, and the right one to the left exactly once. For which minimum

m

is this possible?