Competition problem — Number theory

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Number theory Difficulty 4.7 AIME Find the answer

Let $d$ be a randomly chosen divisor of 2016. Find the expected value of $\frac{d^{2}}{d^{2}+2016}$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Let $ab=2016$ . Then $\frac{a^{2}}{a^{2}+2016}+\frac{b^{2}}{b^{2}+2016}=\frac{a^{2}}{a^{2}+2016}+\frac{\left(\frac{2016}{a}\right)^{2}}{\left(\frac{2016}{a}\right)^{2}+2016}=\frac{a^{2}}{a^{2}+2016}+\frac{2016}{a^{2}+2016}=1$ Thus, every divisor $d$ pairs up with $\frac{2016}{d}$ to get 1, so our desired expected value is $\frac{1}{2}$ .

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.