TW-TST 2002

Find all natural numbers $n$ and nonnegative integers $x_{1},x_{2},...,x_{n}$ such that $\sum_{i=1}^{n}x_{i}^{2}=1+\frac{4}{4n+1}(\sum_{i=1}^{n}x_{i})^{2}$ .

A lattice point $X$ in the plane is said to be visible from the origin $O$ if the line segment $OX$ does not contain any other lattice points. Show that for any positive integer $n$ , there is square $ABCD$ of area $n^{2}$ such that none of the lattice points inside the square is visible from the origin.

Suppose $x,y,,a,b,c,d,e,f$ are real numbers satifying i) $\max{(a,0)}+\max{(b,0)}<x+ay+bz<1+\min{(a,0)}+\min{(b,0)}$ , and ii) $\max{(c,0)}+\max{(d,0)}<cx+y+dz<1+\min{(c,0)}+\min{(d,0)}$ , and iii) $\max{(e,0)}+\max{(f,0)}<ex+fy+z<1+\min{(e,0)}+\min{(f,0)}$ . Prove that $0<x,y,z<1$ .

Let $0<x_{1},x_{2},x_{3},x_{4}\leq\frac{1}{2}$ are real numbers. Prove that $\frac{x_{1}x_{2}x_{3}x_{4}}{(1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})}\leq\frac{x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}}{(1-x_{1})^{4}+(1-x_{2})^{4}+(1-x_{3})^{4}+(1-x_{4})^{4}}$ .

Suppose that the real numbers $a_{1},a_{2},...,a_{2002}$ satisfying $\frac{a_{1}}{2}+\frac{a_{2}}{3}+...+\frac{a_{2002}}{2003}=\frac{4}{3}$ $\frac{a_{1}}{3}+\frac{a_{2}}{4}+...+\frac{a_{2002}}{2004}=\frac{4}{5}$ $...$ $\frac{a_{1}}{2003}+\frac{a_{2}}{2004}+...+\frac{a_{2002}}{4004}=\frac{4}{4005}$ Evaluate the sum $\frac{a_{1}}{3}+\frac{a_{2}}{5}+...+\frac{a_{2002}}{4005}$ .

Let $A,B,C$ be fixed points in the plane , and $D$ be a variable point on the circle $ABC$ , distinct from $A,B,C$ . Let $I_{A},I_{B},I_{C},I_{D}$ be the Simson lines of $A,B,C,D$ with respect to triangles $BCD,ACD,ABD,ABC$ respectively. Find the locus of the intersection points of the four lines $I_{A},I_{B},I_{C},I_{D}$ when point $D$ varies.