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 $x,y\in X_i$.