CMO 1992

Let equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ with real coefficients satisfy $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$ . Suppose that $\lambda$ ( $|\lambda|>1$ ) is a complex root of the equation, prove that $\lambda^{n+1}=1$ .

Given nonnegative real numbers $x_1,x_2,\dots ,x_n$ , let $a=min\{x_1, x_2,\dots ,x_n\}$ . Prove that the following inequality holds: $$\sum^{n}_{i=1}\dfrac{1+x_i}{1+x_{i+1}}\le n+\dfrac{1}{(1+a)^2}\sum^{n}_{i=1}(x_i-a)^2 (x_{n+1}=x_1),$$ and equality occurs if and only if $x_1=x_2=\dots =x_n$ .

Given a $9\times 9$ grid, we assign either $+1$ or $-1$ to each square on the grid. We define an adjustment as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have $81$ products. Next we replace all the squares' values with their corresponding products. Determine if we can make all values in the grid equal to $1$ through finite adjustments.

A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$ . The diagonals $AC$ , $BD$ of $ABCD$ meet at $P$ . Circumcircles of $\triangle ABP$ and $\triangle CDP$ meet at $P$ and $Q$ ( $O,P,Q$ are pairwise distinct). Show that $\angle OQP=90^{\circ}$ .

Let sequence $\{a_1,a_2,\dots \}$ with integer terms satisfy the following conditions:

1. $a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots$ ;
2. $2a_1=a_0+a_2-2$ ;
3. for arbitrary natural number $m$ , there exist $m$ consecutive terms $a_k, a_{k-1}, \dots ,a_{k+m-1}$ among the sequence such that all such $m$ terms are perfect squares. Prove that all terms of the sequence $\{a_1,a_2,\dots \}$ are perfect squares.