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Part of 2022 Caucasus Mathematical Olympiad

Problems(2)

Double of a squares in geometric progression

Source: VII Caucasus Mathematical Olympiad

Positive integers

a

b

c

are given. It is known that

\frac{c}{b}=\frac{b}{a}

, and the number

b^2-a-c+1

is a prime. Prove that

a

c

are double of a squares of positive integers.

algebranumber theorygeometric sequence

In each column different numbers but fixed sum in second row

Source: VII Caucasus Mathematical Olympiad

Given a rectangular table with 2 rows and 100 columns. Dima fills the cells of the first row with numbers 1, 2 or 3. Prove that Alex can fill the cells of the second row with numbers 1, 2, 3 in such a way that the numbers in each column are different and the sum of the numbers in the second row equals 200.