MathDB

2.2

Part of 2016 Saudi Arabia Pre-TST

Problems(2)

black and blue vertices of a regular 20 -gon

Source: 2016 Saudi Arabia Pre-TST Level 4 2.2

9/13/2020
Ten vertices of a regular 2020-gon A1A2....A20A_1A_2....A_{20} are painted black and the other ten vertices are painted blue. Consider the set consisting of diagonal A1A4A_1A_4 and all other diagonals of the same length. 1. Prove that in this set, the number of diagonals with two black endpoints is equal to the number of diagonals with two blue endpoints. 2. Find all possible numbers of the diagonals with two black endpoints.
combinatoricscombinatorial geometryColoring
x_1 =|a- b|, y_1 = |b-c|, z_1 = |c-d|, t_1 = |d -a|, sequence

Source: 2016 Saudi Arabia Pre-TST Level 4+ 2.2

9/13/2020
Given four numbers x,y,z,tx, y, z, t, let (a,b,c,d)(a, b, c, d) be a permutation of (x,y,z,t)(x, y, z, t) and set x1=∣a−b∣x_1 =|a- b|, y1=∣b−c∣y_1 = |b-c|, z1=∣c−d∣z_1 = |c-d|, and t1=∣d−a∣t_1 = |d -a|. From x1,y1,z1,t1x_1, y_1, z_1, t_1, form in the same fashion the numbers x2,y2,z2,t2x_2, y_2, z_2, t_2, and so on. It is known that xn=x,yn=y,zn=z,tn=tx_n = x, y_n = y, z_n = z, t_n = t for some nn. Find all possible values of (x,y,z,t)(x, y, z, t).
algebraSequence