RO-NMO 2019

Let be a point $P$ in the interior of a triangle $ABC$ such that $BP=AC, M$ be the middlepoint of the segment $AP, R$ be the middlepoint of $BC$ and $E$ be the intersection of $BP$ with $AC.$ Prove that the bisector of $\angle BEA$ is perpendicular on $MR$

Find all natural numbers which are the cardinal of a set of nonzero Euclidean vectors whose sum is $0,$ the sum of any two of them is nonzero, and their magnitudes are equal.

Prove that the number of solutions in $\left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right)$ of the parametric equation $$\sqrt{x^2+y+n}+\sqrt{y^2+x+n} = z,$$ is greater than zero and finite, for nay natural number $n.$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x+y)\leq f(x^2+y)$$ for all $x,y$ .

If $a,b,c>0$ then $$\frac{1}{abc}+1\ge3\left(\frac{1}{a^2+b^2+c^2}+\frac{1}{a+b+c}\right)$$

Find all natural numbers $n\ge 4$ that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers $a,b,c$ that verify the equation $$(a-b)^n+(b-c)^n+(c-a)^n=0,$$ represent the vertices of an equilateral triangle in the complex plane.

Let be two nonempty finite sets $A,B$ consisting of nonegative integers. Prove that $$\left\{ f:\mathcal{P} (A)\longrightarrow B| f\left( X\cap Y \right) =\min\left( \left| f(X) \right| ,\left| f(Y) \right| \right) \right\} =\left\{ g:\mathcal{P} (A)\longrightarrow B| f\left( X\cup Y \right) =\max\left( \left| g(X) \right| ,\left| g(Y) \right| \right) \right\}$$ and find the cardinal of these sets.

$\mathcal{P}$ denotes the power of sets.

Let $n \geq 2$ and $A, B \in \mathcal{M}_n(\mathbb{C})$ such that there exists an idempotent matrix $C \in \mathcal{M}_n(\mathbb{C})$ for which $C^*=AB-BA.$ Prove that $(AB-BA)^2=0.$ Note: $X^*$ is the [url = https://en.wikipedia.org/wiki/Adjugate_matrix]adjugate[/url] matrix of $X$ (not the conjugate transpose)

$\textbf{a)}$ Prove that there exists a differentiable function $f:(0, \infty) \to (0, \infty)$ such that $f(f'(x)) = x, \: \forall x>0.$ $\textbf{b)}$ Prove that there is no differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f'(x)) = x, \: \forall x \in \mathbb{R}.$

Let $p$ be a prime number. For any $\sigma \in S_p$ (the permutation group of $\{1,2,...,p \}),$ define the matrix $A_{\sigma}=(a_{ij}) \in \mathcal{M}_p(\mathbb{Z})$ as $a_{ij} = \sigma^{i-1}(j),$ where $\sigma^0$ is the identity permutation and $\sigma^k = \underbrace{\sigma \circ \sigma \circ ... \circ \sigma}_k.$ Prove that $D = \{ |\det A_{\sigma}| : \sigma \in S_p \}$ has at most $1+ (p-2)!$ elements.

Let $a>0$ and $\mathcal{F} = \{f:[0,1] \to \mathbb{R} : f \text{ is concave and } f(0)=1 \}.$ Determine $$\min_{f \in \mathcal{F}} \bigg\{ \left( \int_0^1 f(x)dx\right)^2 - (a+1) \int_0^1 x^{2a}f(x)dx \bigg\}.$$

Let $n \geq 4$ be an even natural number and $G$ be a subgroup of $GL_2(\mathbb{C})$ with $|G| = n.$ Prove that there exists $H \leq G$ such that $\{ I_2 \} \neq H$ and $H \neq G$ such that $XYX^{-1} \in H, \: \forall X \in G$ and $\forall Y \in H$

Let $f:[0, \infty) \to (0, \infty)$ be an increasing function and $g:[0, \infty) \to \mathbb{R}$ be a two times differentiable function such that $g''$ is continuous and $g''(x)+f(x)g(x) = 0, \: \forall x \geq 0.$ $\textbf{a)}$ Provide an example of such functions, with $g \neq 0.$ $\textbf{b)}$ Prove that $g$ is bounded.

Let $n \geq 3$ and $a_1,a_2,...,a_n$ be complex numbers different from $0$ with $|a_i| < 1$ for all $i \in \{1,2,...,n-1 \}.$ If the coefficients of $f = \prod_{i=1}^n (X-a_i)$ are integers, prove that $\textbf{a)}$ The numbers $a_1,a_2,...,a_n$ are distinct. $\textbf{b)}$ If $a_j^2 = a_ia_k,$ then $i=j=k.$