HMMT November

Algebra Difficulty 5.0 AIME Find the answer

Let $\pi$ be a permutation of the numbers from 2 through 2012. Find the largest possible value of $\log _{2} \pi(2) \cdot \log _{3} \pi(3) \cdots \log _{2012} \pi(2012)$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Note that $\begin{aligned} \prod_{i=2}^{2012} \log _{i} \pi(i) & =\prod_{i=2}^{2012} \frac{\log \pi(i)}{\log i} \\ & =\frac{\prod_{i=2}^{2012} \log \pi(i)}{\prod_{i=2}^{2012} \log i} \\ & =1 \end{aligned}$ where the last equality holds since $\pi$ is a permutation of the numbers 2 through 2012.

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