KMO 2001

Let $P$ be a given point inside a convex quadrilateral $O_1O_2O_3O_4$ . For each $i = 1,2,3,4$ , consider the lines $l$ that pass through $P$ and meet the rays $O_iO_{i-1}$ and $O_iO_{i+1}$ (where $O_0 = O_4$ and $O_5 = O_1$ ) at distinct points $A_i(l)$ and $B_i(l)$ , respectively. Denote $f_i(l) = PA_i(l) \cdot PB_i(l)$ . Among all such lines $l$ , let $l_i$ be the one that minimizes $f_i$ . Show that if $l_1 = l_3$ and $l_2 = l_4$ , then the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.

Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$ . Prove that $$(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|$$ and find all cases of equality.

For given positive integers $n$ and $N$ , let $P_n$ be the set of all polynomials $f(x)=a_0+a_1x+\cdots+a_nx^n$ with integer coefficients such that:

[list] (a) $|a_j| \le N$ for $j = 0,1, \cdots ,n$ ; (b) The set $\{ j \mid a_j = N\}$ has at most two elements. [/list]

Find the number of elements of the set $\{f(2N) \mid f(x) \in P_n\}$ .

In a triangle $ABC$ with $\angle B < 45^{\circ}$ , $D$ is a point on $BC$ such that the incenter of $\triangle ABD$ coincides with the circumcenter $O$ of $\triangle ABC$ . Let $P$ be the intersection point of the tangent lines to the circumcircle $\omega$ of $\triangle AOC$ at points $A$ and $C$ . The lines $AD$ and $CO$ meet at $Q$ . The tangent to $\omega$ at $O$ meets $PQ$ at $X$ . Prove that $XO=XD$ .

For a positive integer $n \ge 5$ , let $a_i,b_i (i = 1,2, \cdots ,n)$ be integers satisfying the following two conditions:

[list] (a) The pairs $(a_i,b_i)$ are distinct for $i = 1,2,\cdots,n$ ; (b) $|a_1b_2-a_2b_1| = |a_2b_3-a_3b_2| = \cdots = |a_nb_1-a_1b_n| = 1$ . [/list]

Prove that there exist indices $i,j$ such that $1<|i-j|<n-1$ and $|a_ib_j-a_jb_i|=1$ .