An isolated island has the shape of a circle. Initially there are 9 flowers on the circumference of the island: 5 of the flowers are red and the other 4 are yellow. During the summer 9 new flowers grow on the circumference of the island according to the following rule: between 2 old flowers of the same color a new red flower will grow, between 2 old flowers of different colors, a new yellow flower will grow. During the winter, the old flowers die, and the new survive. The same phenomenon repeats every year.
Is it possible (for some configuration of initial 9 flowers) to get all red flowers after finitely many years?
This one wants a proof. Work it on paper, then read the official solution and mark
yourself. Be honest about it: the record is only any use to you if it is.
Official solution
The answer is "no". Assume that we got all red flowers in the year n for the first time. Then in the year n−1 all the flowers were yellow. We will prove that this is impossible.
Let's change the weird story into the one with the flowers labeled by 1 (instead of red) and 1 (instead of yellow). What really happens is that between two flowers a and b, the new flower will grow and will be labeled by ab. Notice that the initial product of all numbers is 1 , and at the end of each winter the product of the numbers is 1 again, so it will never be equal to -1 hence it is impossible to get the configuration where all the flowers are yellow. This is a contradiction.
Source: NuminaMath-1.5,
licensed Apache-2.0.
Statement and solution reproduced as published; topic, difficulty and ordering added
by this site.