MathDB

3

Part of 2003 Moldova Team Selection Test

Problems(3)

Prove that FI and BC are perpendicular

Source: Moldova IMO-BMO TST 2003, day 2, problem 3.

8/16/2008
The sides [AB] [AB] and [AC] [AC] of the triangle ABC ABC are tangent to the incircle with center I I of the △ABC \triangle ABC at the points M M and N N, respectively. The internal bisectors of the △ABC \triangle ABC drawn form B B and C C intersect the line MN MN at the points P P and Q Q, respectively. Suppose that F F is the intersection point of the lines CP CP and BQ BQ. Prove that FI⊥BC FI\perp BC.
geometryincenterangle bisector
Four points are concyclic

Source: Moldova IMO-BMO TST 2003, day 1, problem 3

8/14/2008
Let ABCD ABCD be a quadrilateral inscribed in a circle of center O O. Let M and N be the midpoints of diagonals AC AC and BD BD, respectively and let P P be the intersection point of the diagonals AC AC and BD BD of the given quadrilateral .It is known that the points O,M,Np O,M,Np are distinct. Prove that the points O,N,A,C O,N,A,C are concyclic if and only if the points O,M,B,D O,M,B,D are concyclic. Proposer: Dorian Croitoru
geometrycircumcirclepower of a pointradical axis
Locus of a point satisfying a given relation

Source: Moldova IMO-BMO TST 2003, day 3, problem 3

8/16/2008
Consider a point M M found in the same plane with the triangle ABC ABC, but not found on any of the lines AB,BC AB,BC and CA CA. Denote by S1,S2 S_1,S_2 and S3 S_3 the areas of the triangles AMB,BMC AMB,BMC and CMA CMA, respectively. Find the locus of M M satisfying the relation: (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)
geometrygeometry unsolved