IMO 1975

Problem 1. Let $x_i,y_i(i=1,2,...,n)$ be real numbers such that

Prove that, if $z_1,z_2,\cdots ,z_n$ is any permutation of $y_1,y_2,\cdots ,y_n$, then

${\sum }_{i=1}^n(x_i-y_i{)}^2\le {\sum }_{i=1}^n(x_i-z_i{)}^2.$

Problem 2. Let $a_1,a_2,a_{3,}\cdots$ be an infinite increasing sequence of positive integers. Prove that for every $p\ge 1$ there are infinitely many $a_m$ which can be written in the form

$a_m=x{a}_p+y{a}_q$

with $x,y$ positive integers and $q>p$.

Problem 3. On the sides of an arbitrary triangle $ABC$, triangles $ABR,BCP,CAQ$ are constructed externally with $\angle CBP=\angle CAQ=45^{\circ },\angle BCP=\angle ACQ=30^{\circ },\angle ABR=\angle BAR=15^{\circ }.$ Prove that $\angle QRP=90^{\circ }$ and $QR=RP$.

Problem 4. 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.)

Problem 5. Determine, with proof, whether or not one can find $1975$ points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.

Problem 6. Find all polynomials $P$, in two variables, with the following properties:

(i) for a positive integer $n$ and all real $t,x,y$

$P(tx,ty)=t^{n}P(x,y)$

(that is, $P$ is homogeneous of degree $n$),

(ii) for all real $a,b,c$,

$P(b+c,a)+P(c+a,b)+P(a+b,c)=0,$

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