MK-MO 1998

Let $ABCDE$ be a convex pentagon with $AB = BC =CA$ and $CD = DE = EC$ . Let $T$ be the centroid of $\vartriangle ABC$ , and $N$ be the midpoint of $AE$ . Compute $\angle NT D$

Prove that the numbers $1,2,...,1998$ cannot be separated into three classes whose sums of elements are divisible by $2000,3999$ , and $5998$ , respectively.

A triangle $ABC$ is given. For every positive numbers $p,q,r$ , let $A',B',C'$ be the points such that $\overrightarrow{BA'} = p\overrightarrow{AB}, \overrightarrow{CB'}=q\overrightarrow{BC}$ , and $\overrightarrow{AC'}=r\overrightarrow{CA}$ . Define $f(p,q,r)$ as the ratio of the area of $\vartriangle A'B'C'$ to that of $\vartriangle ABC$ . Prove that for all positive numbers $x,y,z$ and every positive integer $n$ , $\sum_{k=0}^{n-1}f(x+k,y+k,z+k) = n^3f\left(\frac{x}{n},\frac{y}{n},\frac{z}{n}\right)$ .

If $P$ is the area of a triangle $ABC$ with sides $a,b,c$ , prove that $\frac{ab+bc+ca}{4P} \ge \sqrt3$