MathDB

2004 IberoAmerican

Part of IberoAmerican

Subcontests

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Find the locus of point

In the plane are given a circle with center O O and radius r r and a point A A outside the circle. For any point M M on the circle, let N N be the diametrically opposite point. Find the locus of the circumcenter of triangle AMN AMN when M M describes the circle.

Collinear

Given a scalene triangle ABC ABC. Let A′ A', B′ B', C′ C' be the points where the internal bisectors of the angles CAB CAB, ABC ABC, BCA BCA meet the sides BC BC, CA CA, AB AB, respectively. Let the line BC BC meet the perpendicular bisector of AA′ AA' at A′′ A''. Let the line CA CA meet the perpendicular bisector of BB′ BB' at B′ B'. Let the line AB AB meet the perpendicular bisector of CC′ CC' at C′′ C''. Prove that A′′ A'', B′′ B'' and C′′ C'' are collinear.
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Nice oNE

Let n n and k k be positive integers such as either n n is odd or both n n and k k are even. Prove that exists integers a a and b b such as GCD(a,n) \equal{} GCD(b,n) \equal{} 1 and k \equal{} a \plus{} b

Finite set of points in a plane

Given a set H \mathcal{H} of points in the plane, P P is called an "intersection point of H \mathcal{H}" if distinct points A,B,C,D A,B,C,D exist in H \mathcal{H} such that lines AB AB and CD CD are distinct and intersect in P P. Given a finite set A0 \mathcal{A}_{0} of points in the plane, a sequence of sets is defined as follows: for any j≥0 j\geq0, Aj+1 \mathcal{A}_{j+1} is the union of Aj \mathcal{A}_{j} and the intersection points of Aj \mathcal{A}_{j}. Prove that, if the union of all the sets in the sequence is finite, then Ai=A1 \mathcal{A}_{i}=\mathcal{A}_{1} for any i≥1 i\geq1.
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