IMO 1975

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

In the plane of a triangle $ABC,$ in its exterior $,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$ , $\angle BCP = \angle QCA = 30^{\circ}$ , $\angle ABR = \angle RAB = 15^{\circ}$ .

Prove that a.) $\angle QRP = 90\,^{\circ},$ and b.) $QR = RP.$

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $B.$ ( $A$ and $B$ are written in decimal notation.)

Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?

Determine the polynomials P of two variables so that:

a.) for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$

b.) for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$

c.) $P(1,0) =1.$