MathDB

10

Part of 2017 Hanoi Open Mathematics Competitions

Problems(2)

x^2-2ax+b=0,x^2- 2bx+c=0, x^2-2cx+a=0, integer roots (HOMC 2017 J Q10)

Source:

8/7/2019
Find all non-negative integers a,b,ca, b, c such that the roots of equations: {x2−2ax+b=0x2−2bx+c=0x2−2cx+a=0\begin{cases}x^2 - 2ax + b = 0 \\ x^2- 2bx + c = 0 \\ x^2 - 2cx + a = 0 \end{cases} are non-negative integers.
trinomialIntegerrootsalgebra
8-letter words from {C ,H,M, O}, word sequence (HOMC 2017 S Q10)

Source:

9/8/2019
Consider all words constituted by eight letters from {C,H,M,O}\{C ,H,M, O\}. We arrange the words in an alphabet sequence. Precisely, the first word is CCCCCCCCCCCCCCCC, the second one is CCCCCCCHCCCCCCCH, the third is CCCCCCCMCCCCCCCM, the fourth one is CCCCCCCO,...,CCCCCCCO, ..., and the last word is OOOOOOOOOOOOOOOO. a) Determine the 20172017th word of the sequence? b) What is the position of the word HOMCHOMCHOMCHOMC in the sequence?
combinatoricsWordsSequence