Unique fixed point

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $|f(x)-f(y)|\leq k |x-y|$ for some $k \in (0, 1)$ and all $x, y \in \mathbb{R}$. Show that $f$ has a unique fixed point.

Solution

Convergence of improper integral

Let $a \geq 0$ . Prove that the integral $\displaystyle \int_0^\infty\frac{\sin (x^2+ax)}{x}\, {\rm d}x$.

(Qualifying Exams, Wisconsin / Madison University, 2015)

Solution

Sequence of functions

Give an example of a sequence of functions $f_n:\mathbb{R} \rightarrow \mathbb{R}$ such that each $f_n$

1.  is continuous in $(-\infty, 0)$
2.  is discontinuous in $(0, +\infty)$
3. converges uniformly to a continuous function

Solution

Definite integral and inequality

Let $f$ be a continuous function on $[a, b]$. If for every $x \in [a, b)$ there exists a $y \in (x, b)$ such that $\displaystyle \int_x^y f(t)\, {\rm d}t>0$ then prove that $\displaystyle \int_a^b f(x)\, {\rm d}x>0$.

Solution

Inequality with integrals

Let $f:[0, 1] \rightarrow \mathbb{R}$ be a differentiable function such that:

${\color{gray} \bullet} \;\; f(0)=0$ and
${\color{gray} \bullet} \;\; 0< f'(x) \leq 1$

Prove that:

$$\int_{0}^{x}f^3(t)\, {\rm d}t\leq \left ( \int_{0}^{x}f(t) \, {\rm d}t\right )^2, \;\;  x \in [0, 1]$$

Solution

Vanishing derivative in rational points

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f'(x)=0, \;\; \forall x\in \mathbb{Q}$. Prove that $f$ is constant in $\mathbb{R}$.

Solution

Existence of constant

Let $f$ be a non constant function defined on $[a, b]$ and  $f(a)=f(b)=0$. Prove that there exists a $\xi \in (a, b)$ such that:

$$\bigl|{f'(\xi)}\bigr|>\frac{4}{(b-a)^2}\int_{a}^{b}{|f(x)|\,{\rm d}x}$$

 (Qualifying Exams, Wisconsin-Madison, 2015)

Solution

Convex function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable, convex and bounded function in $\mathbb{R}$. Prove that $f$ is constant.

Solution

Existence of constant

Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function such that
$$\int_0^1 f(x) \, {\rm d}x = \int_0^1 xf(x) \, {\rm d}x \tag{1}$$

Prove that there exists a $c \in (0,1)$ such that $\displaystyle c f(c) = 2 \int_c^0 f(x) \, {\rm d}x$.

(Duong Viet Thong, Nam Dinh University, Vietnam)

Solution (by Paolo Perfetti)

Discontinuous function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined as:
$$f(x)= \left\{\begin{matrix}
 1&,  &x \in \mathbb{Q} \\
 0&,  & x \in \mathbb{R} \setminus \mathbb{Q}
\end{matrix}\right.$$

Prove that $f$ is discontinuous.

Functions preserving convergent series

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that preserve convergent series. (That is the series $\sum \limits_{n=1}^{\infty} f(a_n)$ converges whenever the series $\sum \limits_{n=1}^{\infty} a_n$ converges)

Solution

Rational or irrational?

Let $F(0)=0, \; F(1)=\frac{3}{2}$ and
$$F(n)=\frac{5}{2}F(n-1)-F(n-2), \;\; n \geq 2$$

Examine if the number $\displaystyle \sum_{n=0}^{\infty} \frac{1}{F(2^n)}$ is rational or not.

IMC 2015/ 1st Round/ 3rd problem

Solution

Uniformly continuous function

Let $f:[0, +\infty) \rightarrow \mathbb{R}$ be an integrable and uniformly continuous function. Prove that $\displaystyle \lim_{x \rightarrow +\infty} f(x)=0$.

(Berkley Examinations)

Solution

Zero function from an inequality

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $|f'(x)| \leq |f(x)|, \;\; \forall x \in \mathbb{R}$ and $f(0)=0$. Show that $f$ is the zero function.

Solution

Zero function

Let $f$ be a continuous function on an interval $[a, b]$. If for every function $g$ on the interval $[a, b]$ such $g(a)=g(b)=0$ and $\displaystyle \int_a^b f(x)g(x)\, {\rm d}x=0$ hold , then prove that $f$ is the zero function.

Solution

Proof of AM - GM inequality using concavity

The AM - GM (arithmetic - geometric mean ) inequality is expressed as follows:

$$\sum_{k=1}^{n}a_k \geq n \sqrt[n]{\prod_{k=1}^{n}a_k} \tag{1}$$

One proof was given by the French mathematician  Augustin Luis Cauchy in its lecture book that he had prepared for his students. The proof is based on induction , the so called "back and forth" form of induction. Since then many proofs of this inequality have been discovered. In this topic we give a proof based on the concavity of the $\log $ function.

Proof

Maximum of a function

Let $f:[0, +\infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=\ell+1 $ and $\displaystyle \lim_{x\rightarrow +\infty} f(x)=\ell$. Prove that $f$ has maximum.

Solution

Improper integral involving floor function

Evaluate the integral:

$$\mathcal{L}_1=\int_{0}^{\infty}\left \lfloor x \right \rfloor e^{-x}\, {\rm d}x$$

Solution

Integral of floor function

Evaluate the integral:

$$K_n = \int_{-n}^{n} \left \lfloor x \right \rfloor\, {\rm d}x$$

Solution

Estimation of series

Let $n \geq 2$. Prove that:

$$\sum_{k=1}^{\infty}\left [ 1- \left ( 1-2^{-k} \right )^n \right ] \simeq \ln n $$

Solution