TW-TST 1999

Find all triples $(x,y,z)$ of positive integers such that $(x+1)^{y+1}+1=(x+2)^{z+1}$ .

Let $a_{1},a_{2},...,a_{1999}$ be a sequence of nonnegative integers such that for any $i,j$ with $i+j\leq 1999$ , $a_{i}+a_{j}\leq a_{i+j}\leq a_{i}+a_{j}+1$ . Prove that there exists a real number $x$ such that $a_{n}=[nx]\forall n$ .

There are $1999$ people participating in an exhibition. Among any $50$ people there are two who don't know each other. Prove that there are $41$ people, each of whom knows at most $1958$ people.

Let $P^{*}$ be the set of primes less than $10000$ . Find all possible primes $p\in P^{*}$ such that for each subset $S=\{p_{1},p_{2},...,p_{k}\}$ of $P^{*}$ with $k\geq 2$ and each $p\not\in S$ , there is a $q\in P^{*}-S$ such that $q+1$ divides $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$ .

Let $AD,BE,CF$ be the altitudes of an acute triangle $ABC$ with $AB>AC$ . Line $EF$ meets $BC$ at $P$ , and line through $D$ parallel to $EF$ meets $AC$ and $AB$ at $Q$ and $R$ , respectively. Let $N$ be any poin on side $BC$ such that $\widehat{NQP}+\widehat{NRP}<180^{0}$ . Prove that $BN>CN$ .

There are eight different symbols designed on $n\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\leq k\leq 7)$ the number of shirts containing at least one of the $k$ symbols is even. Determine the value of $n$ .