2

Part of 2024 Assara - South Russian Girl's MO

Problems(2)

p/a+p/b=1, p|a+b

Source: Assara 2024 J2

p

be a prime number. Positive integers numbers

a

b

\frac{p}{a}+\frac{p}{b}=1

a+b

is divisible by

p

. What values can an expression

\frac{a+b}{p}

take? Yu.A.Karpenko

number theoryprime numbers

There is a side that does not more than diameter of incircle in 8-gon

Source: Assara 2024 S2

Prove that in any described

8

-gon there is a side that does not exceed the diameter of the inscribed circle in length. P.A.Kozhevnikov