MathDB

3

Part of 2022 Tuymaada Olympiad

Problems(2)

Tuymaada 2022

Source: Tuymaada 2022 Junior P-3

9/12/2022
Bisectors of a right triangle △ABC\triangle ABC with right angle BB meet at point I.I. The perpendicular to ICIC drawn from BB meets the line IAIA at D;D; the perpendicular to IAIA drawn from BB meets the line ICIC at E.E. Prove that the circumcenter of the triangle △IDE\triangle IDE lies on the line AC.AC. (A. Kuznetsov )
geometry
Tuymaada P3 senior

Source: tuymaada 2022 senior P3

7/3/2022
Is there a colouring of all positive integers in three colours so that for each positive integer the numbers of its divisors of any two colours differ at most by 2?2?
number theorycombinatorics