2

Part of 2021 All-Russian Olympiad

Problems(2)

system of equations to be satisfied

Source: All-Russian 2021/10.2

Find all sets of positive integers

\{x_1, x_2, \dots, x_{20}\}

x_{i+2}^2=lcm(x_{i+1}, x_{i})+lcm(x_{i}, x_{i-1})

i=1, 2, \dots, 20

x_0=x_{20}, x_{21}=x_1, x_{22}=x_2

Tangent lines to graph of polynomial form a closed polygonal chain

Source: All-Russian 2021/11.2

P(x)

be a nonzero polynomial of degree

n>1

with nonnegative coefficients such that function

y=P(x)

is odd. Is that possible thet for some pairwise distinct points

A_{1}, A_{2}, \dots A_{n}

G: y = P(x)

the following conditions hold: tangent to

G

A_{1}

A_{2}

G

A_{2}

A_{3}

\dots

G

A_{n}

A_{1}

algebrapolynomialfunction