5

Part of 2015 All-Russian Olympiad

Problems(3)

Source: All Russian Grade 9 Day 2 P 1

100

integers are arranged in a circle. Each number is greater than the sum of the two subsequent numbers (in a clockwise order). Determine the maximal possible number of positive numbers in such circle. (S.Berlov)

cutting a square into equal figures

Source: All Russian MO 2015, grade 10, problem 5

It is known that a cells square can be cut into

n

equal figures of

k

cells. Prove that it is possible to cut it into

k

equal figures of

n

Flee Jumping on Number Line

Source: All Russian Olympiad 2015 11.5

An immortal flea jumps on whole points of the number line, beginning with

0

. The length of the first jump is

3

5

9

, and so on. The length of

k^{\text{th}}

jump is equal to

2^k + 1

. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?

combinatoricsnumber theoryAll Naturals