5

Part of 2023 Assara - South Russian Girl's MO

Problems(2)

gray cells in 5x5 table

Source: 2023 South Russian Girls MO - Assara Juniors p5

5 \times 5

checkered square, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?

combinatoricsColoring

a + b + c =? if |a - b| + |b - c| + |c - a| = 1

Source: 2023 South Russian Girls MO - Assara Seniors p5

Given three non-negative real numbers

a

b

c

. The sum of the modules of their pairwise differences is equal to

1

|a- b| + |b -c| + |c -a| = 1

. What can the sum

a + b + c