BY-MO 2001

On the Cartesian coordinate plane, the graph of the parabola $y = x^2$ is drawn. Three distinct points $A$ , $B$ , and $C$ are marked on the graph with $A$ lying between $B$ and $C$ . Point $N$ is marked on $BC$ so that $AN$ is parallel to the y-axis. Let $K_1$ and $K_2$ are the areas of triangles $ABN$ and $ACN$ , respectively. Express $AN$ in terms of $K_1$ and $K_2$ .

Prove for postitive $a$ and natural $n$ $$a^n+\frac{1}{a^n}-2 \geq n^2(a+\frac{1}{a}-2)$$

Three distinct points $A$ , $B$ , and $N$ are marked on the line $l$ , with $B$ lying between $A$ and $N$ . For an arbitrary angle $\alpha \in (0,\frac{\pi}{2})$ , points $C$ and $D$ are marked in the plane on the same side of $l$ such that $N$ , $C$ , and $D$ are collinear; $\angle NAD = \angle NBC = \alpha$ ; and $A$ , $B$ , $C$ , and $D$ are concyclic. Find the locus of the intersection points of the diagonals of $ABCD$ as $\alpha$ varies between $0$ and $\frac{\pi}{2}$ .

The problem committee of a mathematical olympiad prepares some variants of the contest. Each variant contains $4$ problems, chosen from a shortlist of $n$ problems, and any two variants have at most one problem in common. (a) If $n = 14$ , determine the largest possible number of variants the problem committee can prepare. (b) Find the smallest value of n such that it is possible to prepare ten variants of the contest.

In the increasing sequence of positive integers $a_1$ , $a_2$ ,. . . , the number $a_k$ is said to be funny if it can be represented as the sum of some other terms (not necessarily distinct) of the sequence. (a) Prove that all but finitely terms of the sequence are funny. (b) Does the result in (a) always hold if the terms of the sequence can be any positive rational numbers?

Let $n$ be a positive integer. Each square of a $(2n-1) \times (2n - 1)$ square board contains an arrow, either pointing up, down,to the left, or to the right. A beetle sits in one of the cells. Each year it creeps from one square in the direction of the arrow in that square, either reaching another square or leaving the board. Each time the beetle moves, the arrow in the square it leaves turns $\frac{\pi}{2}$ clockwise. Prove that the beetle leaves the board in at most $2^{3n-1}(n-1)!-4$ years after it first moves.

The convex quadrilateral $ABCD$ is inscribed in the circle $S_1$ . Let $O$ be the intersection of $AC$ and $BD$ . Circle $S_2$ passes through $D$ and $O$ , intersecting $AD$ and $CD$ at $M$ and $N$ , respectively. Lines $OM$ and $AB$ intersect at $R$ , lines $ON$ and $BC$ intersect at $T$ , and $R$ and $T$ lie on the same side of line $BD$ as $A$ . Prove that $O$ , $R$ , $T$ , and $B$ are concyclic.

There are $n$ aborigines on an island. Any two of them are either friends or enemies. One day, the chieftain orders that all citizens (including himself) make and wear a necklace with zero or more stones so that: (i) given a pair of friends, there exists a color such that each has a stone of that color; (ii) given a pair of enemies,there does not exist a color such that each a stone of that color.

(a) Prove that the aborigines can carry out the chieftain's order. (b) What is the minimum number of colors of stones required for the aborigines to carry out the chieftain's order?