MathDB

P4

Part of 2023 Bangladesh Mathematical Olympiad

Problems(2)

Balls and buckets

Source: BdMO 2023 Secondary National P4

2/12/2023
20232023 balls are divided into several buckets such that no bucket contains more than 9999 balls. We can remove balls from any bucket or remove an entire bucket, as many times as we want. Prove that we can remove them in such a way that each of the remaining buckets will have an equal number of balls and the total number of remaining balls will be at least 100100.
combinatorics
Orthocenter of variable triangles

Source: BdMO 2023 Higher Secondary National P4

2/12/2023
Let ABCDABCD be an isosceles trapezium inscribed in circle ω\omega, such that AB∣∣CDAB||CD. Let PP be a point on the circle ω\omega. Let H1H_1 and H2H_2 be the orthocenters of triangles PADPAD and PBCPBC respectively. Prove that the length of H1H2H_1H_2 remains constant, when PP varies on the circle.
geometryorthocenter