Fun of Thinking — Math Problems from Around the World

Define

\rho(x, y) = \left|e^x - y\right| - y\left|x - \ln y\right|, \qquad x \in \mathbb{R},\ y \in \mathbb{R}^+.

1. For $a > 0$ , let $f(x) = \rho(x, a)$ . Find the number of zeros of $f(x)$ . 2. For $0 < a < b$ , let $F(x) = \rho(x, a) - \rho(x, b)$ . Find the minimum value of $F(x)$ . 3. Denote the minimum from part 2 by $T(a, b)$ . If $\{a_n\}$ is a strictly increasing sequence of positive numbers, prove that for every positive integer $n$ ,

\sum_{i=1}^{n} T\left(a_i, a_{i+1}\right) < \left(a_{n+1} - a_1\right)\ln 2.

Difficulty	9/10
Topics	functions, calculus, Absolute Value, Monotonicity, Telescoping, inequality

A Rho Distance Function

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