MathDB

11.6

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(3)

all triangles DPQ by moving point M are similar to each other

Source: IV Soros Olympiad 1997-98 R2 11.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/1/2024
It is known that the bisector of the angle ∠ADC\angle ADC of the inscribed quadrilateral ABCDABCD passes through the center of the circle inscribed in the triangle ABCABC. Let MM be an arbitrary point of the arc ABCABC of the circle circumscribed around ABCDABCD. Denote by PP and QQ the centers of the circles inscribed in the triangles ABMABM and BCMBCM. Prove that all triangles DPQDPQ obtained by moving point MM are similar to each other. Find the angle ∠PDQ\angle PDQ and ratio BP:PQBP : PQ if ∠BAC=α\angle BAC = \alpha, ∠BCA=β\angle BCA = \beta
geometrysimilar trianglescyclic quadrilateral
max no of acute, 6 points (IV Soros Olympiad 1997-98 Correspondence 11.6)

Source:

6/1/2024
There are 66 points marked on the plane. Find the greatest possible number of acute triangles with vertices at the marked points.
geometrycombinatoricscombinatorial geometry
shortest path on surface of rectangular parallelepiped

Source: IV Soros Olympiad 1997-98 R3 11. 6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/2/2024
On the planet Brick, which has the shape of a rectangular parallelepiped with edges of 11 km,2 2 km and 44 km, the Little Prince built a house in the center of the largest face. What is the distance from the house to the most remote point on the planet? (The distance between two points on the surface of a planet is defined as the length of the shortest path along the surface connecting these points.)
geometry3D geometrygeometric inequality