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.
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.
Find the minimum for all \(\alpha\), \(\beta\) of the maximum of the function \(y (x) = | \cos x + \alpha \cos 2x + \beta \cos 3x |\).
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.
Is there a line on the coordinate plane relative to which the graph of the function \(y = 2^x\) is symmetric?
The function \(f (x)\) for each real value of \(x\in (-\infty, + \infty)\) satisfies the equality \(f (x) + (x + 1/2) \times f (1 - x) = 1\).
a) Find \(f (0)\) and \(f (1)\). b) Find all such functions \(f (x)\).
Aladdin visited all of the points on the equator, moving to the east, then to the west, and sometimes instantly moving to the diametrically opposite point on Earth. Prove that there was a period of time during which the difference in distances traversed by Aladdin to the east and to the west was not less than half the length of the equator.
Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?
The numbers \(a\) and \(b\) are such that the first equation of the system \[\begin{aligned} \sin x + a & = bx \\ \cos x &= b \end{aligned}\] has exactly two solutions. Prove that the system has at least one solution.