International Mathematical Olympiad 1982

The function $f(n)$ is defined for all positive integers $n$ and takes on non-negative integer values. Also, for all $m, n$

$$f(m + n) - f(m) - f(n) = 0 \text{ or } 1$$

$$f(2) = 0, f(3) > 0, \text{ and } f(9999) = 3333.$$

Determine $f(1982)$.

A non-isosceles triangle $A_1A_2A_3$ is given with sides $a_1, a_2, a_3$ ($a_i$ is the side opposite $A_i$). For all $i = 1, 2, 3, M_i$ is the midpoint of side $a_i$, and $T_i$ is the point where the incircle touches side $a_i$. Denote by $S_i$ the reflection of $T_i$ in the interior bisector of angle $A_i$. Prove that the lines $M_1S_1$, $M_2S_2$, and $M_3S_3$ are concurrent.

Consider the infinite sequences $\{x_n\}$ of positive real numbers with the following properties:

$$x_0 = 1, \text{ and for all } i \geq 0, x_{i+1} \leq x_i.$$

(a) Prove that for every such sequence, there is an $n \geq 1$ such that

$$\frac{x_0^2}{x_1} + \frac{x_1^2}{x_2} + \cdots + \frac{x_{n-1}^2}{x_n} \geq 3.999.$$

(b) Find such a sequence for which

$$\frac{x_0^2}{x_1} + \frac{x_1^2}{x_2} + \cdots + \frac{x_{n-1}^2}{x_n} < 4.$$

Prove that if $n$ is a positive integer such that the equation

$$x^3 - 3xy^2 + y^3 = n$$

has a solution in integers $(x, y)$, then it has at least three such solutions.

Show that the equation has no solutions in integers when $n = 2891$.

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by the inner points $M$ and $N$, respectively, so that

$$\frac{AM}{AC} = \frac{CN}{CE} = r.$$

Determine $r$ if $B$, $M$, and $N$ are collinear.

Let $S$ be a square with sides of length 100, and let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1, A_1A_2, \ldots, A_{n-1}A_n$ with $A_0 \neq A_n$. Suppose that for every point $P$ of the boundary of $S$ there is a point of $L$ at a distance from $P$ not greater than $1/2$. Prove that there are two points $X$ and $Y$ in $L$ such that the distance between $X$ and $Y$ is not greater than 1, and the length of that part of $L$ which lies between $X$ and $Y$ is not smaller than 198.