MathDB

2

Part of 2018 Thailand TSTST

Problems(3)

Draw 9 horizontal and 9 vertical line through a square

Source: 2017 Thailand October Camp 1.2

2/25/2022
99 horizontal and 99 vertical lines are drawn through a square. Prove that it is possible to select 2020 rectangles so that the sides of each rectangle is a segment of one of the given lines (including the sides of the square), and for any two of the 2020 rectangles, it is possible to cover one of them with the other (rotations are allowed).
combinatoricslinesconstruction
Three stick wth integer length

Source: 2017 Thailand October Camp 2.2

2/25/2022
There are three sticks, each of which has an integer length which is at least nn; the sum of their lengths is n(n+1)/2n(n + 1)/2. Prove that it is possible to break the sticks (possibly several times) so that the resulting sticks have length 1,2,…,n1, 2,\dots, n.
Note: a stick of length a+ba + b can be broken into sticks of lengths aa and bb.
combinatorics
ME = BC wanted, AE = MD, AN = AM, <BAC = 135^o, circumcircle

Source: 2017 Thailand October Camp 3.2

10/15/2020
In triangle △ABC\vartriangle ABC, ∠BAC=135o\angle BAC = 135^o. MM is the midpoint of BCBC, and N≠MN \ne M is on BCBC such that AN=AMAN = AM. The line AMAM meets the circumcircle of △ABC\vartriangle ABC at DD. Point EE is chosen on segment ANAN such that AE=MDAE = MD. Show that ME=BCME = BC.
geometryequal segmentscircumcircle