MathNet 2018-part-05 — 181 problems (EN–VI)

Question 1

Find all $x$ that satisfy
$$
\log_2(x^2 + 4) - \log_2 x + x^2 - 4x + 2 = 0.
$$

Question 2

Prove that the number
$$
\sqrt[n]{\sqrt{2018} + \sqrt{2017}} + \sqrt[n]{\sqrt{2018} - \sqrt{2017}}
$$
is not rational for $n \ge 2$.

Question 3

Consider real numbers $a, b, c$ satisfying $1 \lt b \le c^2 \le a^{10}$, and
$$
\log_a b + 2 \log_b c + 5 \log_c a = 12.
$$
Show that
$$
2 \log_a c + 5 \log_b c + 10 \log_b a \ge 21.
$$

Question 4

For a fixed integer $n \ge 2$, determine complex numbers $z$ that satisfy the relations:
$$
a)\ z^n + z^{n-1} + \dots + z^2 + |z| = n;
$$
$$
b)\ |z|^{n-1} + |z|^{n-2} + \dots + |z|^2 + z = n z^n.
$$

Prove that for every integer $n \ge 2$ one can find matrices $A_1, A_2, \dots, A_n \in \mathcal{M}_2(\mathbb{R})$, having non-zero entries, such that
$$
A_1^{-1} + A_2^{-1} + \dots + A_n^{-1} = (A_1 + A_2 + \dots + A_n)^{-1}.
$$

Question 6

Define the set $M = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathcal{M}_2(\mathbb{C}) \mid ab = cd \right\}$.

a) Give examples of $A \in M$ having the property $A^{2017} \in M$ and $A^{2019} \in M$, but $A^{2018} \notin M$.

b) Prove that if for an $A \in M$ there is $k \ge 1$ such that $A^k \in M$, $A^{k+1} \in M$ and $A^{k+2} \in M$, then $A^n \in M$, for all integers $n \ge 1$.

Question 7

The sequence $(a_n)_{n \ge 1}$ has the properties that $a_n \gt 1$ and $a_{n+1}^2 \ge a_n a_{n+2}$, for $n \ge 1$. Show that the sequence $(x_n)_{n \ge 1}$, defined by $x_n = \log_{a_n} a_{n+1}$, for $n \ge 1$, has a finite limit. Find it.

Question 8

For real $a \lt b$, and $f : (a, b) \to \mathbb{R}$ a function such that the functions defined by $g : (a, b) \to \mathbb{R}$, $g(x) = (x-a)f(x)$, and $h : (a, b) \to \mathbb{R}$, $h(x) = (x-b)f(x)$ are non-decreasing, prove that $f$ is continuous.

Vladimir Cerbu and Sorin Rădulescu

Question 9

Let $\mathcal{F}$ be the set of all continuous functions $f: [0, 1] \to \mathbb{R}$, satisfying
$$ \max_{0 \le x \le 1} |f(x)| = 1, \text{ and let } I: \mathcal{F} \to \mathbb{R}, $$
$$
I(f) = \int_{0}^{1} f(x) \, dx - f(0) + f(1).
$$

Question 10

Let $p$ be an integer greater or equal to $2$ and let $(M, \cdot)$ be a finite semi-group, such that $a^p \neq a$, for all $a \in M \setminus \{e\}$, where $e$ is the null element in $M$. Prove that $(M, \cdot)$ is a group.

Traian Preda

Question 11

Prove that a continuous function $f: \mathbb{R} \to \mathbb{R}$ is non-decreasing if and only if
$$
(c - b) \int_{a}^{b} f(x) \, dx \leq (b - a) \int_{b}^{c} f(x) \, dx,
$$
for all $a \lt b \lt c$.

Question 12

Consider two integers $n$ and $q$, $n \ge 2$, $q \ge 2$, and $q \ne 1 \pmod 4$. Let $K$ be a finite field having exactly $q$ elements. Show that for any $a$ from $K$, there is $x$ and $y$ in $K$, such that $a = x^2 + y^2$. (It is known that any finite field is commutative.)

Question 13

Show that if $n \geq 2$ is an integer, then there exist invertible matrices $A_{1}, A_{2}, \ldots, A_{n} \in \mathcal{M}_{2}(\mathbb{R})$, with non-zero entries, such that $A_{1}^{-1}+A_{2}^{-1}+\ldots+A_{n}^{-1}=\left(A_{1}+A_{2}+\ldots+A_{n}\right)^{-1}$.

Question 14

Consider the set $M=\left\{\left.\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in \mathcal{M}_2(\mathbb{C}) \right\vert\, a b=c d\right\}$.
a) Give an example of a matrix $A \in M$ such that $A^{2017} \in M$ and $A^{2019} \in M$, but $A^{2018} \notin M$.
b) Show that if $A \in M$ and there exists an integer $k \geq 1$ such that $A^{k} \in M, A^{k+1} \in M$ and $A^{k+2} \in M$, then $A^{n} \in M$ for every integer $n \geq 1$.

Question 15

Let $a \lt b$ be real numbers and $f : (a, b) \rightarrow \mathbb{R}$ a function such that the functions $g : (a, b) \rightarrow \mathbb{R}$, $g(x) = (x - a) f(x)$ and $h : (a, b) \rightarrow \mathbb{R}$, $h(x) = (x - b) f(x)$ are increasing (non-decreasing). Show that the function $f$ is continuous on $(a, b)$.