MathDB

1

Part of 2007 CentroAmerican

Problems(2)

n divides the year of the nth Olympiad

Source:

6/12/2007
The Central American Olympiad is an annual competition. The ninth Olympiad is held in 2007. Find all the positive integers nn such that nn divides the number of the year in which the nn-th Olympiad takes place.
Synonymous words

Source: Central American Olympiad 2007, Problem 4

6/12/2007
In a remote island, a language in which every word can be written using only the letters aa, bb, cc, dd, ee, ff, gg is spoken. Let's say two words are synonymous if we can transform one into the other according to the following rules: i) Change a letter by another two in the following way: a→bc, b→cd, c→de, d→ef, e→fg, f→ga, g→aba \rightarrow bc,\ b \rightarrow cd,\ c \rightarrow de,\ d \rightarrow ef,\ e \rightarrow fg,\ f\rightarrow ga,\ g\rightarrow ab ii) If a letter is between other two equal letters, these can be removed. For example, dfd→fdfd \rightarrow f. Show that all words in this language are synonymous.
algorithm