MathDB

9.5

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

area wanted, 3 angles 120^o (V Soros Olympiad 1998-99 Round 1 9.5)

Source:

5/25/2024
In triangle ABCABC, ∠BAC=60o\angle BAC= 60^o. Point PP is taken inside the triangle so that angles ∠APB=∠BPC=∠CPA=120o\angle APB=\angle BPC= \angle CP A=120^o. It is known that AP=aAP = a. Find the area of triangle BPCBPC.
geometryareas
locus of vertice of isosceles triangle (V Soros Olympiad 1998-99 Round 2 9.5)

Source:

5/25/2024
An angle with vertex AA is given on the plane. Points KK and PP are selected on its sides so that AK+AP=aAK + AP = a, where aa is a given segment. Let MM be a point on the plane such that the triangle KPMKPM is isosceles with the base KPKP and the angle at the vertex MM equal to the given angle. Find the locus of points MM (as KK and PP move).
geometryLocusisosceles
computational geo with trapezoid (V Soros Olympiad 1998-99 Round 3 9.5)

Source:

5/25/2024
In the trapezoid ABCDABCD with bases BC=aBC = a, AD=bAD = b, the equality holds: ∠BAC+∠ACD=180o\angle BAC + \angle ACD = 180^o. The straight line ACAC intersects the common tangents to the circumcircles of the triangles ABCABC and ACDACD at the points Find PQPQ.
geometrytrapezoid