IMO 1982

Problem 1. 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 $f(1982)$.

Problem 2. A non-isosceles triangle $A_1{A}_2{A}_3$ 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 $M_1,S_1,M_2{S}_2,$ and $M_3{S}_3$ are concurrent.

Problem 3. Consider the infinite sequences $\{x_n\}$ of positive real numbers with the following properties:

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

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

(b) Find such a sequence for which

Problem 4. Prove that if $n$ is a positive integer such that the equation

$x^3-3x{y}^2+y^3=n$

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 $n=2891$.

Problem 5. 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.

Problem 6. 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 $A_0{A}_1,A_1{A}_2,\cdots ,A_{n-1}{A}_n$ with $A_0\ne A_n$ . 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 $198$.