CWMO 2002

Find all positive integers $n$ such that $n^4-4n^3+22n^2-36n+18$ is a perfect square.

Let $O$ be the circumcenter of acute triangle $ABC$ . Point $P$ is in the interior of triangle $AOB$ . Let $D,E,F$ be the projections of $P$ on the sides $BC,CA,AB$ , respectively. Prove that the parallelogram consisting of $FE$ and $FD$ as its adjacent sides lies inside triangle $ABC$ .

In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $x^4 + px^3 + qx^2 + rx + s = 0$ . Find the minimum of square areas.

Let $n$ be a positive integer, let the sets $A_{1},A_{2},\cdots,A_{n + 1}$ be non-empty subsets of the set $\{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $\{1,2,\cdots,n + 1\}$ : $\{i_{1},i_{2},\cdots,i_{k}\}$ and $\{j_{1},j_{2},\cdots,j_{m}\}$ such that $A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} = A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$ .

Given a trapezoid $ABCD$ with $AD\parallel BC, E$ is a moving point on the side $AB,$ let $O_{1},O_{2}$ be the circumcenters of triangles $AED,BEC$ , respectively. Prove that the length of $O_{1}O_{2}$ is a constant value.

Given a positive integer $n$ , find all integers $(a_{1},a_{2},\cdots,a_{n})$ satisfying the following conditions: $(1): a_{1}+a_{2}+\cdots+a_{n}\ge n^2;$ $(2): a_{1}^2+a_{2}^2+\cdots+a_{n}^2\le n^3+1.$

Assume that $\alpha$ and $\beta$ are two roots of the equation: $x^2-x-1=0$ . Let $a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$ , $n=1, 2, \cdots$ . (1) Prove that for any positive integer $n$ , we have $a_{n+2}=a_{n+1}+a_n$ . (2) Find all positive integers $a$ and $b$ , $a<b$ , satisfying $b \mid a_n-2na^n$ for any positive integer $n$ .

Assume that $S=(a_1, a_2, \cdots, a_n)$ consists of $0$ and $1$ and is the longest sequence of number, which satisfies the following condition: Every two sections of successive $5$ terms in the sequence of numbers $S$ are different, i.e., for arbitrary $1\le i<j\le n-4$ , $(a_i, a_{i+1}, a_{i+2}, a_{i+3}, a_{i+4})$ and $(a_j, a_{j+1}, a_{j+2}, a_{j+3}, a_{j+4})$ are different. Prove that the first four terms and the last four terms in the sequence are the same.