In a plane, $3n$ points ($n>1$) are given, no three collinear, and every pair has distance at most $1$. Prove that one can construct $n$ pairwise disjoint triangles such that (1) every given point is a vertex of exactly one triangle and (2) their total area is less than $\tfrac12$. Two triangles are disjoint if they have no common point in their interiors or on their boundaries.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
For each positive integer $n$, let $f(n)$ be the greatest integer for which $2^{f(n)}$ divides
$$a_n=\sum_{i=0}^{\lfloor(n-1)/2\rfloor}\binom n{2i+1}3^i.$$
Find all positive integers $n$ with $f(n)=1996$.
Notation disclosure: the source prints $\binom{2i+1}{n}$, with the selection above the total, consistently with its other reversed binomial notation and its binomial expansion. Here it is explicitly normalized to $\binom n{2i+1}$. Under the usual top-is-total reading of the printed symbol the sum would be zero and $f(n)$ would not be defined.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Let $A,B,C$ be three noncollinear points. For each point $M$ in their plane, reflect $M$ in line $AB$ to obtain $M_1$, reflect $M_1$ in line $BC$ to obtain $M_2$, and reflect $M_2$ in line $CA$ to obtain $M'$. Determine all points $M$ for which $MM'$ is least, and call this least distance $d$. Prove that any other order of the three reflections, producing $M''$, has the same least possible distance $MM''=d$.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
A number of students are to be invited to a meeting. Each is unacquainted with at least $56$ other invited students, and every pair who are unacquainted have at least one invited student acquainted with both. Can the number invited be $65$?
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Find all real numbers $a$ such that the sequence defined by
$$x_0=\sqrt{1996},\qquad x_{n+1}=\frac a{1+x_n^2}\quad(n=0,1,2,\ldots)$$
has a finite limit as $n\to\infty$.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.