In the plane, let $S$ consist of $n\ge3$ distinct points satisfying: (1) the distance between any two points of $S$ is at most 1; (2) each point $A$ has exactly two neighbors, meaning points of $S$ at distance 1 from $A$; (3) if $A',A''$ are the neighbors of $A$ and $B',B''$ those of $B$, then $\angle A' AA''=\angle B' BB''$ for any $A,B\in S$. Does such a set exist for $n=1991$? For $n=2000$? Explain.
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Let $a_1,\ldots,a_n$ be positive real numbers, where $n>2$ and $a_1\ne a_n$, forming a nondecreasing sequence ($a_k\le a_{k+1}$ for $1\le k\lt n$) or a nonincreasing sequence ($a_k\ge a_{k+1}$ for $1\le k\lt n$). Let $x,y$ be positive real numbers such that $$\frac xy\ge\frac{a_1-a_2}{a_1-a_n}.$$ Prove that, using cyclic indices $a_{n+1}=a_1$, $a_{n+2}=a_2$, $$\sum_{k=1}^n\frac{a_k}{a_{k+1}x+a_{k+2}y}\ge\frac n{x+y}.$$
The positive real sequence $(x_n)$ is defined by $$x_1=1,\quad x_2=9,\quad x_3=9,\quad x_4=1,\qquad x_{n+4}=\sqrt[4]{x_nx_{n+1}x_{n+2}x_{n+3}}\quad(n\ge1).$$ Prove that it has a limit and find the limit.
Let $T$ be any tetrahedron such that every edge has length at most 1 and every face is a right triangle. Let $s(T)$ be the sum of the squares of the areas of its four faces. Find the largest value of $s(T)$.
For each natural number $n$, define $f(1)=1$ and, if $n>1$ has prime factorization $n=p_1^{a_1}\cdots p_k^{a_k}$ with distinct primes $p_i$ and positive integer exponents $a_i$, set $$f(n)=1+a_1p_1+\cdots+a_kp_k.$$ Let $f_s(n)$ denote the result of applying $f$ exactly $s$ times. Prove that for every given natural number $a$, there is a natural number $s_0$ such that $f_s(a)+f_{s-1}(a)$ is independent of the integer $s>s_0$. Here natural numbers in the domain are positive, as in the supplied definition.
Let $X$ be a set of $2n$ distinct real numbers, where $n\ge3$. Let $K$ be a set of ordered pairs $(x,y)$ with $x,y\in X$ and $x\ne y$, satisfying: (1) if $(x,y)\in K$, then $(y,x)\notin K$; (2) each element $x\in X$ occurs in at most 19 pairs of $K$. Prove that $X$ can be partitioned into five nonempty pairwise disjoint subsets $X_1,\ldots,X_5$ such that, for each $i=1,\ldots,5$, at most $3n$ pairs $(x,y)\in K$ have both $x,y\in X_i$.
Let an odd integer $k\gt 1$ be given. For each positive integer $n$, define $f(n)$ to be the largest nonnegative integer such that $k^n-1$ is divisible by $2^{f(n)}$. Determine a formula for $f(n)$ in terms of $k$ and $n$.
Let $Oxyz$ be a right trihedral angle, and let fixed points $A,B,C$ lie respectively on $Ox,Oy,Oz$. A variable sphere $(E)$ always passes through $A,B,C$ and meets the edges $Ox,Oy,Oz$ again at $A',B',C'$, respectively. Let $M,M'$ be the centroids of triangles $ABC,A'B'C'$, respectively. As $(E)$ varies, on what figure does the midpoint $S$ of $MM'$ move?
There are $1991$ students standing in a circle, facing inward, to play the following counting game. Each student says one number in turn clockwise, starting at a student $A$. The numbers said are $1,2,3$, repeated in that order. A student who says $2$ or $3$ must leave the circle immediately. The last remaining student receives a prize. Counting clockwise from $A$, who first says $1$, which starting position should a student choose to win?
Triangle $ABC$ has centroid $G$ and is inscribed in a circle of radius $R$. The medians from $A,B,C$, extended, meet the circle again at $D,E,F$, respectively. Prove that
$$\frac3R\le\frac1{GD}+\frac1{GE}+\frac1{GF}\le\sqrt3\left(\frac1{AB}+\frac1{BC}+\frac1{CA}\right).$$