MathDB

3

Part of 2023 Girls in Mathematics Tournament

Problems(2)

FM bisects the angle AFD?

Source: 2023 Girls in Mathematics Tournament- Level A- Problem 3

10/29/2023
Let ABCABC an acute triangle and DD and EE the feet of heights by AA and BB, respectively, and let MM be the midpoint of ACAC. The circle that passes through DD and BB and is tangent to BEBE in BB intersects the line BMBM in F,F≠BF, F\neq B. Show that FMFM is the angle bisector of ∠AFD\angle AFD.
geometry
Set of positive integers and coloring points in red

Source: 2023 Girls in Mathematics Tournament- Level B, Problem 3

10/29/2023
Let SS be a set not empty of positive integers and ABAB a segment with, initially, only points AA and BB colored by red. An operation consists of choosing two distinct points X,YX, Y colored already by red and n∈Sn\in S an integer, and painting in red the nn points A1,A2,...,AnA_1, A_2,..., A_n of segment XYXY such that XA1=A1A2=A2A3=...=An−1An=AnYXA_1= A_1A_2= A_2A_3=...= A_{n-1}A_n= A_nY and XA1<XA2<...<XAnXA_1<XA_2<...<XA_n. Find the least positive integer mm such exists a subset SS of {1,2,..,m}\{1,2,.., m\} such that, after a finite number of operations, we can paint in red the point KK in the segment ABAB defined by AKKB=27092022\frac{AK}{KB}= \frac{2709}{2022}. Also, find the number of such subsets for such a value of mm.
combinatorics