BG-MO 1987

Let $f(x)=x^n+a_1x^{n-1}+\ldots+a_n~(n\ge3)$ be a polynomial with real coefficients and $n$ real roots, such that $\frac{a_{n-1}}{a_n}>n+1$ . Prove that if $a_{n-2}=0$ , then at least one root of $f(x)$ lies in the open interval $\left(-\frac12,\frac1{n+1}\right)$ .

Let there be given a polygon $P$ which is mapped onto itself by two rotations: $\rho_1$ with center $O_1$ and angle $\omega_1$ , and $\rho_2$ with center $O_2$ and angle $\omega_2~(0<\omega_i<2\pi)$ . Show that the ratio $\frac{\omega_1}{\omega_2}$ is rational.

Let $MABCD$ be a pyramid with the square $ABCD$ as the base, in which $MA=MD$ , $MA^2+AB^2=MB^2$ and the area of $\triangle ADM$ is equal to $1$ . Determine the radius of the largest ball that is contained in the given pyramid.

The sequence $(x_n)_{n\in\mathbb N}$ is defined by $x_1=x_2=1$ , $x_{n+2}=14x_{n+1}-x_n-4$ for each $n\in\mathbb N$ . Prove that all terms of this sequence are perfect squares.

Let $E$ be a point on the median $AD$ of a triangle $ABC$ , and $F$ be the projection of $E$ onto $BC$ . From a point $M$ on $EF$ the perpendiculars $MN$ to $AC$ and $MP$ to $AB$ are drawn. Prove that if the points $N,E,P$ lie on a line, then $M$ lies on the bisector of $\angle BAC$ .

Let $\Delta$ be the set of all triangles inscribed in a given circle, with angles whose measures are integer numbers of degrees different than $45^\circ,90^\circ$ and $135^\circ$ . For each triangle $T\in\Delta$ , $f(T)$ denotes the triangle with vertices at the second intersection points of the altitudes of $T$ with the circle.

(a) Prove that there exists a natural number $n$ such that for every triangle $T\in\Delta$ , among the triangles $T,f(T),\ldots,f^n(T)$ (where $f^0(T)=T$ and $f^k(T)=f(f^{k-1}(T))$ ) at least two are equal. (b) Find the smallest $n$ with the property from (a).