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Problem 1656

National Olympiad, first round

Combinatorics Difficulty 6.3 Prove it Estonian Mathematical Olympiad · Estonia

Every sound in a certain language can be either long or short. A sound is classified either as a vowel or as a consonant. Every word consists of exactly two sounds (without repetitions) and satisfies the following conditions.

1) Every word contains a short sound.
2) Words beginning with a vowel contain a long sound.
3) Words beginning with a consonant or ending with a vowel have a short sound at the end.

All sequences of two distinct sounds satisfying these conditions are words. The written language optimisation committee has decided to denote each sound with a different letter. However, they are considering two possibilities for denoting length. The first proposes denoting vowels with single letters and consonants with single or double letters based on length. The second instead proposes denoting consonants with single letters and vowels with single or double letters based on length. Is it possible to determine the lengths of sounds in all words from writing: a) in the case of the first proposal; b) in the case of the second proposal?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

a) If the word consists of two vowels, then based on the rule 3) the second of them is short and based on the rule 2) the first of them is long. If the word begins with a vowel and ends with a consonant, then the length of the second sound is determined uniquely by its writing and the length of first sound must be of the opposite length based on 1) and 2). If the word begins with a consonant, the length of first sound is determined uniquely by its writing and the second sound is short based on 3). Therefore, all writings of words in the language have unique pronunciation.

b) The rules allow both: a word consisting of two short consonants, and a word starting with a long consonant and ending with a short consonant. According to the proposal, the first sound of these words have identical writing and cannot be uniquely determined.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.