MathDB

8

Part of 2023 Assara - South Russian Girl's MO

Problems(2)

n-gon's perimeter > 2n-gon's perimeter, perimeters ACE=ABCDEF

Source: 2023 South Russian Girls MO - Assara Juniors p8

2/28/2024
a) Given a convex hexagon ABCDEFABCDEF, which has a center of symmetry. Prove that the perimeter of triangle ACEACE is greater than half the perimeter of hexagon ABCDEFABCDEF.
b) Given a convex (2n)(2n)-gon PP having a center of symmetry, its vertices are colored alternately red and blue. Let QQ be an nn-gon with red vertices. Is it possible to say that the perimeter of QQ is certainly greater than half the perimeter PP? Solve the problem for n=4n = 4 and n=5n = 5.
geometryperimeter
sequence of letters ASSARA...

Source: 2023 South Russian Girls MO - Assara Seniors p8

3/1/2024
The girl continues the sequence of letters ASSARA...ASSARA... , adding one of the three letters AA, RR or SS. When adding the next letter, the girl makes sure that no two written sevens of consecutive letters coincide. At some point it turned out that it was impossible to add a new letter according to these rules. What letter could be written last?
combinatorics