Let $a \ge 2$ be a positive integer. Prove that the following statements are equivalent:
a) One can find positive integers $b, c$, such that $a^2 = b^2 + c^2$.
b) One can find a positive integer $d$, such that the equations $x^2 - a x + d = 0$ and $x^2 - a x - d = 0$ have integer roots.
Let $a$, $b$, $c \in \mathbb{C}^*$, be distinct complex numbers with equal moduli, such that $a^2 + b^2 + c^2 + ab + ac + bc = 0$. Prove that $a$, $b$, $c$ are the complex coordinates of the vertices of a triangle which is either right angled or equilateral.
Let $\alpha$ and $\beta$ be real numbers. Find, with proof, the maximal value of the expression
$$
|\alpha x + \beta y| + |\alpha x - \beta y|,
$$
in each of the following cases:
a) $x, y \in \mathbb{R}$, such that $|x| \le 1$ and $|y| \le 1$;
b) $x, y \in \mathbb{C}$, such that $|x| \le 1$ and $|y| \le 1$.
a) Prove that there exist non-periodical functions $f : \mathbb{R} \to \mathbb{R}$ such that
$$
f(x + 1) + f(x - 1) = \sqrt{5}f(x),
$$
for all $x \in \mathbb{R}$.
b) Prove that any function $g : \mathbb{R} \to \mathbb{R}$ such that
$$
g(x + 1) + g(x - 1) = \sqrt{3}g(x),
$$
for all $x \in \mathbb{R}$, is periodical.
Determine the continuous functions $f : \mathbb{R} \to \mathbb{R}$ having the property that
$$
f\left(x + \frac{1}{n}\right) \le f(x) + \frac{1}{n}, \text{ for all } x \in \mathbb{R} \text{ and } n \in \mathbb{Z}^*.
$$
Let $I \subset \mathbb{R}$ be an open interval and $f, g : I \to \mathbb{R}$ two functions that satisfy
$$
\frac{f(x) - g(y)}{x - y} + |x - y| \ge 0, \text{ for any } x, y \in I, x \ne y.
$$
i) Deduce that $f$ and $g$ are non-decreasing.
ii) Find $f, g : \mathbb{R} \to \mathbb{R}$, $f \ne g$, having the above property.
A ring $(A, +, \cdot)$ has property (P) if $A$ is finite and the multiplicative group of units is isomorphic to a non-trivial subgroup of the additive group $(A, +)$. Show that:
(a) the number of elements of a ring having property (P) is even;
(b) there are $n$-element rings having property (P) for infinitely many positive integers $n$.
Let $f: [0, 1] \to [0, 1]$ be an increasing function and let
$$
a_n = \int_0^1 \frac{1 + (f(x))^n}{1 + (f(x))^{n+1}} \, dx, \quad n = 1, 2, 3, \dots
$$
Prove that the sequence $(a_n)_{n \ge 1}$ is convergent and evaluate its limit.
Let $A, B, C, D \in \mathcal{M}_{n}(\mathbb{C})$, $n \geq 2$ and $k \in \mathbb{R}$ be such that $A C + k B D = I_{n}$ and $A D = B C$. Prove that $C A + k D B = I_{n}$ and $D A = C B$.
Let $a, b, c \in \mathbb{C}^*$ be distinct and have the same modulus, such that
$$
a^{2}+b^{2}+c^{2}+a b+a c+b c=0.
$$
Prove that $a, b, c$ are the affixes of the vertices of a right-angled or equilateral triangle.
a) Prove that there exist non-periodic functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy the equality
$$
f(x+1)+f(x-1)=\sqrt{5} f(x)
$$
for all $x \in \mathbb{R}$.
b) Prove that any function $g: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies the equality
$$
g(x+1)+g(x-1)=\sqrt{3} g(x)
$$
for all $x \in \mathbb{R}$, is periodic.
A ring $(A,+, \cdot)$ has property (P) if $A$ is finite and the multiplicative group of its invertible elements is isomorphic to a subgroup of the additive group $(A,+)$ different from $\{0\}$. Show that:
a. If a ring has property $(\mathrm{P})$, then its number of elements is even.
b. For infinitely many natural numbers $n$, there exist rings with exactly $n$ elements which have property $(\mathrm{P})$.
Let $p$ be an odd prime number and let $G$ be a group with exactly $p+1$ elements. Show that if $p$ divides the number of automorphisms of $G$, then $p \equiv 3 \pmod 4$.
Let $f:[0,1] \rightarrow [0,1]$ be an increasing function and let
$$
a_{n} = \int_{0}^{1} \frac{1 + (f(x))^{n}}{1 + (f(x))^{n+1}} \, \mathrm{d}x, \quad n \in \mathbb{N}^{*}.
$$
Show that the sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ is convergent and compute its limit.