6

Part of 2022 Assara - South Russian Girl's MO

Problems(2)

bw in 9x9 table

Source: 2022 South Russian Girls MO - Assara Juniors p6

The cells of the

9 \times 9

table are colored black and white. It turned out, that there were

k

rows, in each of which there are more black cells than white ones white, and there were

k

columns, each of which contained more than black. At what highest

k

is this possible?

combinatoricsColoring

sum a_ia_{i+2} = 0 if a_i =\sqrt2 a_{i+2} - \sqrt3 a_{i+1}

Source: 2022 South Russian Girls MO - Assara Seniors p6

2022

numbers arranged in a circle

a_1, a_2, . . ,a_{2022}

. It turned out that for any three consecutive

a_i

a_{i+1}

a_{i+2}

a_i =\sqrt2 a_{i+2} - \sqrt3 a_{i+1}

\sum^{2022}_{i=1} a_ia_{i+2} = 0

, if we know that

a_{2023} = a_1

a_{2024} = a_2

algebraSequencerecurrence relation