Problems

In a square with side length 1 there is a broken line, which does not self-intersect, whose length is no less than 200. Prove that there is a straight line parallel to one of the sides of the square that intersects the broken line at a point no less than 101 units along the line.

A square $ABCD$ contains 5 points. Prove that the distance between some pair of these points does not exceed $\frac{1}{2}AC$.

A white plane is arbitrarily sprinkled with black ink. Prove that for any positive $l$ there exists a line segment of length $l$ with both ends of the same colour.

The numbers $a_1,a_2,\dots ,a_{1985}$ are the numbers $1,2,\dots ,1985$ rearranged in some order. Each number $a_k$ is multiplied by its number $k$, and then the largest number is chosen among the resulting 1985 products. Prove that it is not less than ${993}^2$.

The product of 1986 natural numbers has exactly 1985 different prime factors. Prove that either one of these natural numbers, or the product of several of them, is the square of a natural number.

The product of a group of 48 natural numbers has exactly 10 prime factors. Prove that the product of some four of the numbers in the group will always give a square number.

In March 2015 a teacher ran 11 sessions of a maths club. Prove that if no sessions were run on Saturdays or Sundays there must have been three days in a row where a session of the club did not take place. The 1st March 2015 was a Sunday.

Prove that from any 27 different natural numbers less than 100, two numbers that are not coprime can be chosen.

7 different digits are given. Prove that for any natural number $n$ there is a pair of these digits, the sum of which ends in the same digit as the number.

In a group of seven boys, everyone has at least three brothers who are in that group. Prove that all seven are brothers.