CMO 1998

Let $ABC$ be a non-obtuse triangle satisfying $AB>AC$ and $\angle B=45^{\circ}$ . The circumcentre $O$ and incentre $I$ of triangle $ABC$ are such that $\sqrt{2}\ OI=AB-AC$ . Find the value of $\sin A$ .

Given a positive integer $n>1$ , determine with proof if there exist $2n$ pairwise different positive integers $a_1,\ldots ,a_n,b_1,\ldots b_n$ such that $a_1+\ldots +a_n=b_1+\ldots +b_n$ and $$n-1>\sum_{i=1}^{n}\frac{a_i-b_i}{a_i+b_i}>n-1-\frac{1}{1998}.$$

Let $S=\{1,2,\ldots ,98\}$ . Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$ -element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.

Find all natural numbers $n>3$ , such that $2^{2000}$ is divisible by $1+C^1_n+C^2_n+C^3_n$ .

Let $D$ be a point inside acute triangle $ABC$ satisfying the condition $$DA\cdot DB\cdot AB+DB\cdot DC\cdot BC+DC\cdot DA\cdot CA=AB\cdot BC\cdot CA.$$ Determine (with proof) the geometric position of point $D$ .

Let $x_1,x_2,\ldots ,x_n$ be real numbers, where $n\ge 2$ , satisfying $\sum_{i=1}^{n}x^2_i+ \sum_{i=1}^{n-1}x_ix_{i+1}=1$ . For each $k$ , find the maximal value of $|x_k|$ .