1

Part of 2022 Assara - South Russian Girl's MO

Problems(2)

blondes, brunettes, redheads and brown-haired women

Source: 2022 South Russian Girls MO - Assara Juniors p1

Blondes, brunettes, redheads and brown-haired women participate in the Olympiad. There are twice as many redheads as brown-haired. Blondes and redheads make up a quarter of the total number of participants, and Brown-haired and Blondes one fifth part. Prove that the number of Brunettes is divisible by

7

a, b and c are perfect squares

Source: 2022 South Russian Girls MO - Assara Seniors p1

Given three natural numbers

a

b

c

. It turned out that they are coprime together. And their least common multiple and their product are perfect squares. Prove that

a

b

c

are perfect squares.

number theoryPerfect SquaresPerfect Square