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.
Find all real numbers $x,y$ such that
$$\begin{cases}\sqrt{3x}\left(1+\dfrac1{x+y}\right)=2,\\[3pt]\sqrt{7y}\left(1-\dfrac1{x+y}\right)=4\sqrt2.\end{cases}$$
Let $SABC$ be a tetrahedron. Triangles $DAB,EBC,FCA$ in the plane of $ABC$ are similar respectively to $SAB,SBC,SCA$. Let $O$ be the circumcenter of $DEF$. Suppose $K$ is the “circumsphere of $SABC$ opposite $O$” (tangent to the planes $SAB,SBC,SCA,ABC$, on the opposite side of $ABC$ from $S$, but on the same side of $SAB$ as $C$, the same side of $SBC$ as $A$, and the same side of $SCA$ as $B$). Prove that $K$ touches the plane $ABC$ at $O$.
Let $n$ be a positive integer and let $k$ be a positive integer not exceeding $n$. Determine the number of ordered $k$-element tuples $(a_1,a_2,\ldots,a_n)$ satisfying:
(a) All $a_i$ are distinct and belong to $\{1,2,\ldots,n\}$.
(b) $a_r\gt a_s$ for some $r\lt s$.
(c) $a_r$ has the same parity as $r$ for some $r$.
Determine all functions $f$ on the positive integers, with positive integer values, satisfying
$$f(n)+f(n+1)=f(n+2)f(n+3)-1996$$
for every positive integer $n$.
Triangle $ABC$ has $BC=1$ and angle $A=x$. Let $f(x)$ be the least possible distance between its incenter and centroid. Determine $f(x)$. What is the largest value of $f(x)$ for $60^\circ\lt x\lt 180^\circ$?