Joy of Mathematics

$\varphi$ summation

October 15, 2021 / Leave a comment

Let $\varphi$ denote the Euler’s function. Prove that

$\sum_{k=1}^{\infty} \varphi(k) \left \lfloor \frac{n}{k} \right \rfloor = \frac{n \left ( n+1 \right )}{2}$

Solution

The key idea is to rewrite the floor as a sum involving divisors. Hence,

$\begin{align*} \sum_{k=1}^{\infty} \varphi(k) \left \lfloor \frac{n}{k} \right \rfloor &= \sum_{k=1}^{\infty} \varphi(k) \sum_{\substack{m \leq n \\ k \mid n}} 1 \\ &= \sum_{k=1}^{\infty} \sum_{\substack{m \leq n \\ k \mid n}} \varphi(k) \\ &= \sum_{m=1}^{n} \sum_{k \mid m} \varphi(k) \\ &= \sum_{m=1}^{n} m \\ &= \frac{n \left ( n+1 \right )}{2} \end{align*}$

Equal segments

October 14, 2021 / Leave a comment

Let $f(x) = \frac{1}{x}$ and let $(\varepsilon)$ be a line that intersects $\mathcal{C}_f$ at the points $\mathrm{A}$ , $\Delta$ and the axis at the points $\mathrm{B}$ and $\Gamma$ .

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Prove that $\mathrm{AB}= \Gamma \Delta$ .

Solution

Let $(\varepsilon):y= \alpha x + \beta$ with $\alpha \neq 0$ . Then,

$\alpha x + \beta = \frac{1}{x} \Leftrightarrow \alpha x^2 + \beta x - 1 = 0 \Leftrightarrow \left\{\begin{matrix} x_1 + x_2 &= & - \dfrac{\beta}{\alpha} \\\\ x_1 x_2 & = & - \dfrac{1}{\alpha} \end{matrix}\right.$

Hence,

$\begin{align*} \overrightarrow{\mathrm{BA}} &= \left ( x_1, \frac{1}{x_1} - \beta \right ) \\ &= \left ( - \frac{\beta}{\alpha} -x_2, - \frac{1}{x_2} \right ) \\ &= \overrightarrow{\Delta \Gamma} \end{align*}$

Integral inequality (I)

August 20, 2021 / Leave a comment

Let $f:[0, 1] \rightarrow \mathbb{R}$ be a differentiable and convex function such that $f(0)=0$ and $f'(1)= 2$ . Prove that $\int_{0}^{1} f(x) \, \mathrm{d}x < 1$ .

Symmetry

July 24, 2021 / Leave a comment

Let $f, g : \mathbb{R} \rightarrow \mathbb{R}$ be continuous functions. If $\mathcal{C}_f$ is symmetric around the line $x=\frac{\alpha + \beta}{2}$ then prove that: