The function $f(n)$ is defined for all positive integers $n$ and takes on nonnegative integer values. Also, for all $m,n$,
$f(m+n)-f(m)-f(n)=0$ or $1$;
$f(2)=0$, $f(3)>0$, and $f(9999)=3333$.
Determine $f(1982)$.
A nonisosceles triangle $A_1A_2A_3$ is given with sides $a_1,a_2,a_3$ ($a_i$ is the side opposite to $A_i$). For all $i=1,2,3$, $M_i$ is the midpoint of side $a_i$, $T_i$ is the point where the incircle touches side $a_i$, and the reflection of $T_i$ in the interior bisector of $A_i$ yields the point $S_i$. Prove that the lines $M_1S_1,M_2S_2,M_3S_3$ are concurrent.
Consider the infinite sequences $\{x_n\}$ of positive real numbers with the following properties:
$x_0=1$ and for all $i\ge0$, $x_{i+1}\le x_i$.
(a) Prove that for every such sequence there is an $n\ge1$ such that $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots+\frac{x_{n-1}^2}{x_n}\ge3.999$.
(b) Find such a sequence for which $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots+\frac{x_{n-1}^2}{x_n}\lt 4$ for all $n$.
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 solution 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 $B,M,N$ are collinear.
Let $S$ be a square with sides of length 100 and let $L$ be a path within $S$ that does not meet itself and that is composed of linear segments $A_0A_1,A_1A_2,\ldots,A_{n-1}A_n$ 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 $\frac12$. 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 the part of $L$ that lies between $X$ and $Y$ is not smaller than 198.