4

Part of 2016 Saint Petersburg Mathematical Olympiad

Problems(3)

n points with a constant speed each move around a circle

Source: St. Petersburg MO 2016 11.4

N> 4

points move around the circle, each with a constant speed. For Any four of them have a moment in time when they all meet. Prove that is the moment when all the points meet.

combinatorial geometrycombinatoricsspeedcirclepoints

Transitioning between admissible coloring

Source: Saint Petersburg 2016

The cells of a square

100 \times 100

table are colored in one of two colors, black or white. A coloring is called admissible if for any row or column, the number

b

of black colored cells satisfies

50 \le b \le 60

. It is permitted to recolor a cell if by doing so the resulting configuration is still admissible. Prove that one can transition from any admissible coloring of the board to any other using a sequence of valid recoloring operations.

combinatoricsColoring

Existence of consecutive with special property

Source: St.Petersburg 2016 Grade 9 P4

Two different prime numbers

p

q

differ in less than

2

times. Prove that exists two consecutive natural numbers, such that largest prime divisor of first number is

p

, and largest prime divisor of second number is

q