Prove that the following statement is true for $n=3$ and for $n=5$, and false for all other $n\gt 2$:
For any real numbers $a_1,a_2,\ldots,a_n$,
$$(a_1-a_2)(a_1-a_3)\cdots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots(a_2-a_n)+\ldots$$
$$+(a_n-a_1)(a_n-a_2)\cdots(a_n-a_{n-1})\ge0.$$
Given a convex polyhedron $P_1$ with $9$ vertices $A_1,\ldots,A_9$, let us denote by $P_2,P_3,\ldots,P_9$ the images of $P_1$ under the translations mapping the vertex $A_1$ to $A_2,A_3,\ldots,A_9$, respectively. Prove that among the polyhedra $P_1,\ldots,P_9$ at least two have a common interior point.
Given a tetrahedron $ABCD$ all of whose faces are acute-angled triangles, set
$$\sigma=\angle DAB+\angle BCD-\angle ABC-\angle CDA.$$
Consider all closed broken lines $XYZTX$ whose vertices $X,Y,Z,T$ lie in the interior of segments $AB,BC,CD,DA$ respectively. Prove that:
(a) if $\sigma\ne0$, then there is no broken line $XYZT$ of minimal length;
(b) if $\sigma=0$, then there are infinitely many such broken lines of minimal length. That length equals $2AC\sin(\alpha/2)$, where
$$\alpha=\angle BAC+\angle CAD+\angle DAB.$$
Prove that for every natural number $m\ge1$ there exists a finite set $S_m$ of points in the plane satisfying the following condition: If $A$ is any point in $S_m$, then there are exactly $m$ points in $S_m$ whose distance to $A$ equals $1$.
Consider the $n\times n$ array of nonnegative integers
$$\begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\ldots&a_{nn}\end{pmatrix},$$
with the following property: If an element $a_{ij}$ is zero, then the sum of the elements of the $i$th row and the $j$th column is greater than or equal to $n$. Prove that the sum of all the elements is greater than or equal to $\ge\frac12n^2$.