TW-TST 1997

Let $a$ be rational and $b,c,d$ are real numbers, and let $f: \mathbb{R}\to [-1.1]$ be a function satisfying $f(x+a+b)-f(x+b)=c[x+2a+[x]-2[x+a]-[b]]+d$ for all $x$ . Show that $f$ is periodic.

Given a line segment $AB$ in the plane, find all possible points $C$ such that in the triangle $ABC$ , the altitude from $A$ and the median from $B$ have the same length.

Let $n>2$ be an integer. Suppose that $a_{1},a_{2},...,a_{n}$ are real numbers such that $k_{i}=\frac{a_{i-1}+a_{i+1}}{a_{i}}$ is a positive integer for all $i$ (Here $a_{0}=a_{n},a_{n+1}=a_{1}$ ). Prove that $2n\leq a_{1}+a_{2}+...+a_{n}\leq 3n$ .

Let $k=2^{2^{n}}+1$ for some $n\in\mathbb{N}$ . Show that $k$ is prime iff $k|3^{\frac{k-1}{2}}+1$ .

Let $ABCD$ is a tetrahedron. Show that a)If $AB=CD,AC=DB,AD=BC$ then triangles $ABC,ABD,ACD,BCD$ are acute. b)If the triangles $ABC,ABD,ACD,BCD$ have the same area , then $AB=CD,AC=DB,AD=BC$ .

Show that every number of the form $2^{p}3^{q}$ , where $p,q$ are nonnegative integers, divides some number of the form $a_{2k}10^{2k}+a_{2k-2}10^{2k-2}+...+a_{2}10^{2}+a_{0}$ , where $a_{2i}\in\{1,2,...,9\}$

Find all positive integers $k$ for which there exists a function $f: \mathbb{N}\to\mathbb{Z}$ satisfying $f(1997)=1998$ and $f(ab)=f(a)+f(b)+kf(\gcd{(a,b)})\forall a,b$ .

Let $O$ be the circumcenter and $R$ be the circumradius of an acute triangle $ABC$ . Let $AO$ meet the circumcircle of $OBC$ again at $D$ , $BO$ meet the circumcircle of $OCA$ again at $E$ , and $CO$ meet the circumcircle of $OAB$ again at $F$ . Show that $OD.OE.OF\geq 8R^{3}$ .

For $n\geq k\geq 3$ , let $X=\{1,2,...,n\}$ and let $F_{k}$ a the family of $k$ -element subsets of $X$ , any two of which have at most $k-2$ elements in common. Show that there exists a subset $M_{k}$ of $X$ with at least $[\log_{2}{n}]+1$ elements containing no subset in $F_{k}$ .