Let $x_1\ge x_2\ge\cdots\ge x_n$ and $y_1\ge y_2\ge\cdots\ge y_n$ be two $n$-tuples of numbers. Prove that
$$\sum_{i=1}^n(x_i-y_i)^2\le\sum_{i=1}^n(x_i-z_i)^2$$
is true when $z_1,z_2,\ldots,z_n$ denote $y_1,y_2,\ldots,y_n$ taken in another order.
Let $a_1,a_2,a_3,\ldots$ be any infinite increasing sequence of positive integers. (For every integer $i\gt 0$, $a_{i+1}\gt a_i$.) Prove that there are infinitely many $m$ for which positive integers $x,y,h,k$ can be found such that $0\lt h\lt k\lt m$ and $a_m=xa_h+ya_k$.
On the sides of an arbitrary triangle $ABC$, triangles $BPC$, $CQA$, and $ARB$ are externally erected such that
$$\angle PBC=\angle CAQ=45^\circ,$$
$$\angle BCP=\angle QCA=30^\circ,$$
$$\angle ABR=\angle BAR=15^\circ.$$
Prove that $\angle QRP=90^\circ$ and $QR=RP$.
Let $A$ be the sum of the digits of the number $4444^{4444}$ and $B$ the sum of the digits of the number $A$. Find the sum of the digits of the number $B$.
Is it possible to plot 1975 points on a circle with radius 1 so that the distance between any two of them is a rational number (distances have to be measured by chords)?
The function $f(x,y)$ is a homogeneous polynomial of the $n$th degree in $x$ and $y$. If $f(1,0)=1$ and for all $a,b,c$,
$$f(a+b,c)+f(b+c,a)+f(c+a,b)=0,$$
prove that $f(x,y)=(x-2y)(x+y)^{n-1}$.