Question bank IMO 1993

Let $n\gt 1$ be an integer and let $f(x)=x^n+5x^{n-1}+3$. Prove that there do not exist polynomials $g(x),h(x)$, each having integer coefficients and degree at least one, such that $f(x)=g(x)h(x)$.

$A,B,C,D$ are four points in the plane, with $C,D$ on the same side of the line $AB$, such that $AC\cdot BD=AD\cdot BC$ and $\angle ADB=90^\circ+\angle ACB$. Find the ratio
$$\frac{AB\cdot CD}{AC\cdot BD},$$
and prove that circles $ACD,BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicular.)

On an infinite chessboard, a solitaire game is played as follows: At the start, we have $n^2$ pieces occupying $n^2$ squares that form a square of side $n$. The only allowed move is a jump horizontally or vertically over an occupied square to an unoccupied one, and the piece that has been jumped over is removed. For what positive integers $n$ can the game end with only one piece remaining on the board?

For three points $A,B,C$ in the plane we define $m(ABC)$ to be the smallest length of the three altitudes of the triangle $ABC$, where in the case of $A,B,C$ collinear, $m(ABC)=0$. Let $A,B,C$ be given points in the plane. Prove that for any point $X$ in the plane,
$$m(ABC)\le m(ABX)+m(AXC)+m(XBC).$$

Let $\mathbb N=\{1,2,3,\ldots\}$. Determine whether there exists a strictly increasing function $f:\mathbb N\to\mathbb N$ with the following properties:
$$f(1)=2;\tag{1}$$
$$f(f(n))=f(n)+n\qquad(n\in\mathbb N).\tag{2}$$

Let $n$ be an integer greater than $1$. In a circular arrangement of $n$ lamps $L_0,\ldots,L_{n-1}$, each one of that can be either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $\mathrm{Step}_0,\mathrm{Step}_1,\ldots$. If $L_{j-1}$ ($j$ is taken mod $n$) is ON, then $\mathrm{Step}_j$ changes the status of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the status of any of the other lamps. If $L_{j-1}$ is OFF, then $\mathrm{Step}_j$ does not change anything at all. Show that:
(a) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again.
(b) If $n$ has the form $2^k$, then all lamps are ON after $n^2-1$ steps.
(c) If $n$ has the form $2^k+1$, then all lamps are ON after $n^2-n+1$ steps.