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Problem of the Day #18: Language Rules April 6, 2011

Posted by Arjun in : potd , add a comment

In some weird language (which for whatever reason uses a $26$ letter Latin alphabet), there are certain rules about the order in which words are spelled. The letter ‘e’ always begins every word, and cannot be placed anywhere else. The second letter is always a consonant. Consonants can be followed by other consonants (except itself) and the vowels ‘a’ and ‘o.’ The letter ‘o’ must be followed by three consonants. The letters ‘a’, ‘i’, and ‘u’ can be followed by any letter. The letter ‘i’ always ends a word, but can appear anywhere in the word (while working within already stated rules). The letter ‘y’ is considered a consonant and not a vowel. How many possible $12$-letter words are there that contain less than $5$ consonants?

Problem of the Day #10: Rational Logs March 29, 2011

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Given that $a$, $b$, $x$, and $y$ are positive integers, and $a+b = 10$, find the number of ordered pairs $(a,b)$ such that: $$ a^{5\log_2x} = b^{\log_{32}y}$$ where the quotient of $\log x$ and $\log y$ is rational.

An Integral March 25, 2011

Posted by Arjun in : calculus , 3 comments

Find $$\int^{\frac{\pi}{2}}_0 \frac{1}{\sqrt{\tan{x}}} \,dx$$