BG-MO 1982

Let $n$ unit circles be given on a plane. Prove that on one of the circles there is an arc of length at least $\frac{2\pi}n$ not intersecting any other circle.

In a regular $2n$ -gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$ . If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$ .

If $x_1,x_2,\ldots,x_n$ are arbitrary numbers from the interval $[0,2]$ , prove that $$\sum_{i=1}^n\sum_{j=1}^n|x_i-x_j|\le n^2$$ When is the equality attained?

Find all values of parameters $a,b$ for which the polynomial $$x^4+(2a+1)x^3+(a-1)^2x^2+bx+4$$ can be written as a product of two monic quadratic polynomials $\Phi(x)$ and $\Psi(x)$ , such that the equation $\Psi(x)=0$ has two distinct roots $\alpha,\beta$ which satisfy $\Phi(\alpha)=\beta$ and $\Phi(\beta)=\alpha$ .

Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$ .