KZ-MO 1999

Prove that for any real numbers $a_1, a_2, \dots, a_ {100}$ there exists a real number $b$ such that all numbers $a_i + b$ ( $1 \leq i \leq 100$ ) are irrational.

Prove that for any odd $n$ there exists a unique polynomial $P (x)$ $n$ -th degree satisfying the equation $P \left (x- \frac {1} {x} \right) = x ^ n- \frac {1} {x ^ n}.$ Is this true for any natural number $n$ ?

The circle inscribed in the triangle $ABC$ , with center $O$ , touches the sides $AB$ and $BC$ at the points $C_1$ and $A_1$ , respectively. The lines $CO$ and $AO$ intersect the line $C_1A_1$ at the points $K$ and $L$ . $M$ is the midpoint of $AC$ and $\angle ABC = 60^\circ$ . Prove that $KLM$ is a regular triangle.

Seven dwarfs live in one house and each has its own hat. One morning one day, two dwarfs inadvertently exchanged hats. At any time, any three gnomes can sit down at the round table and exchange hats clockwise. Is it possible that by evening all the gnomes will be with their hats.

For real numbers $x_1, x_2, \dots, x_n$ and $y_1, y_2, \dots, y_n$ , the inequalities hold $x_1 \geq x_2 \geq \ldots \geq x_n> 0$ and $$y_1 \geq x_1, ~ y_1y_2 \geq x_1x_2, ~ \dots, ~ y_1y_2 \dots y_n \geq x_1x_2 \dots x_n.$$ Prove that $ny_1 + (n-1) y_2 + \dots + y_n \geq x_1 + 2x_2 + \dots + nx_n$ .

In a sequence of natural numbers $a_1$ , $a_2$ , $\dots$ , $a_ {1999}$ , $a_n-a_ {n-1} -a_ {n-2}$ is divisible by $100 (3 \leq n \leq 1999)$ . It is known that $a_1 = 19$ and $a_2 = 99$ . Find the remainder of $a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2$ by $8$ .

On a sphere with radius $1$ , a point $P$ is given. Three mutually perpendicular the rays emanating from the point $P$ intersect the sphere at the points $A$ , $B$ and $C$ . Prove that all such possible $ABC$ planes pass through fixed point, and find the maximum possible area of the triangle $ABC$

Let ${{a} _ {1}}, {{a} _ {2}}, \ldots, {{a} _ {n}}$ be permutation of numbers $1,2, \ldots, n$ , where $n \geq 2$ . Find the maximum value of the sum $$S (n) = | {{a} _ {1}} - {{a} _ {2}} | + | {{a} _ {2}} - {{a} _ {3}} | + \cdots + | {{a} _ {n-1}} - {{a} _ {n}} |.$$