MathDB

1

Part of 2017 Thailand TSTST

Problems(2)

concurrent wanted, midpoints, incenters related

Source: 2016 Thailand October Camp 1.1

10/15/2020
In △ABC,D,E,F\vartriangle ABC, D, E, F are the midpoints of AB,BC,CAAB, BC, CA respectively. Denote by OA,OB,OCO_A, O_B, O_C the incenters of △ADF,△BED,△CFE\vartriangle ADF, \vartriangle BED, \vartriangle CFE respectively. Prove that OAE,OBF,OCDO_AE, O_BF, O_CD are concurrent.
geometryincenterconcurrencyconcurrentmidpoints
3 Problems Algebra, Combinaroric, and Number Theory

Source: 2016 Thailand October Camp 2.1

2/26/2022
1.1 Let f(A)f(A) denote the difference between the maximum value and the minimum value of a set AA. Find the sum of f(A)f(A) as AA ranges over the subsets of {1,2,…,n}\{1, 2, \dots, n\}.
1.2 All cells of an 8×88 × 8 board are initially white. A move consists of flipping the color (white to black or vice versa) of cells in a 1×31\times 3 or 3×13\times 1 rectangle. Determine whether there is a finite sequence of moves resulting in the state where all 6464 cells are black.
1.3 Prove that for all positive integers mm, there exists a positive integer nn such that the set {n,n+1,n+2,…,3n}\{n, n + 1, n + 2, \dots , 3n\} contains exactly mm perfect squares.
combinatoricstableColoringSetsnumber theoryalgebra