MathDB

2.1

Part of 2016 Saudi Arabia Pre-TST

Problems(2)

x_1 = |x - y|, y_1 = | y -z|, z_1 = |z- x| , sequence

Source: 2016 Saudi Arabia Pre-TST Level 4 2.1

9/13/2020
Given three numbers x,y,zx, y, z, and set x1=∣x−y∣,y1=∣y−z∣,z1=∣z−x∣x_1 = |x - y|, y_1 = | y -z|, z_1 = |z- x|. From x1,y1,z1x_1, y_1, z_1, form in the same fashion the numbers x2,y2,z2x_2, y_2, z_2, and so on. It is known that xn=x,yn=y,zn=zx_n = x, y_n = y, z_n = z for some nn. Find all possible values of (x,y,z)(x, y, z).
algebraSequence
permutaion of 1,2,..., n with any 2 adjacent is 2015 or 2016

Source: 2016 Saudi Arabia Pre-TST Level 4+ 2.1

9/13/2020
1) Prove that there are infinitely many positive integers nn such that there exists a permutation of 1,2,3,...,n1, 2, 3, . . . , n with the property that the difference between any two adjacent numbers is equal to either 20152015 or 20162016. 2) Let kk be a positive integer. Is the statement in 1) still true if we replace the numbers 20152015 and 20162016 by kk and k+2016k + 2016, respectively?
combinatoricspermutation