MathDB

1

Part of 2024 Girls in Mathematics Tournament

Problems(2)

words that contain some palindrome

Source: 2024 Girls in Mathematics Tournament, Level A, Problem 1

10/26/2024
A word is a sequence of capital letters of our alphabet (that is, there are 26 possible letters). A word is called palindrome if has at least two letters and is spelled the same forward and backward. For example, the words "ARARA" e "NOON" are palindromes, but the words "ESMERALDA" and "A" are not palindromes. We say that a word xx contains a word yy if there are consecutive letters of xx that together form the word yy. For example, the word "ARARA" contains the word "RARA" and also the word "ARARA", but doesn't contain the word "ARRA". Compute the number of words of 14-letter that contain some palindrome.
combinatorics
a^2-bc=b^2-ac=c^2-ab=a^3+b^3+c^3; possible values of (a+b+c)

Source: 2024 Girls in Mathematics Tournament, Level B, Problem 1

10/26/2024
The nonzero real numbers a,b,ca,b,c are such that: a2−bc=b2−ac=c2−ab=a3+b3+c3a^2-bc= b^2-ac= c^2-ab= a^3+b^3+c^3. Compute the possible values of a+b+ca+b+c.
algebra