RO-NMO 2017

Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.

Let be a square $ABCD,$ a point $E$ on $AB,$ a point $N$ on $CD,$ points $F,M$ on $BC,$ name $P$ the intersection of $AN$ with $DE,$ and name $Q$ the intersection of $AM$ with $EF.$ If the triangles $AMN$ and $DEF$ are equilateral, prove that $PQ=FM.$

Let be two natural numbers $n$ and $a.$ a) Prove that there exists an $n\text{-tuplet}$ of natural numbers $\left( a_1,a_2,\ldots ,a_n\right)$ that satisfy the following equality. $$1+\frac{1}{a} =\prod_{i=1}^n \left( 1+\frac{1}{a_i} \right)$$ b) Show that there exist only finitely such $n\text{-tuplets} .$

Let be two natural numbers $b>a>0$ and a function $f:\mathbb{R}\longrightarrow\mathbb{R}$ having the following property. $$f\left( x^2+ay\right)\ge f\left( x^2+by\right) , \forall x,y\in\mathbb{R}$$ a) Show that $f(s)\le f(0)\le f(t) ,$ for any real numbers $s<0<t.$ b) Prove that $f$ is constant on the interval $(0,\infty ) .$ c) Give an example of a non-monotone such function.

Solve in the set of real numbers the equation $a^{[ x ]} +\log_a\{ x \} =x ,$ where $a$ is a real number from the interval $(0,1).$ $[]$ and $\{\}$ denote the floor, respectively, the fractional part.

A function $f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q}$ has the following property: $$f(xy)=f(x)+f(y), x,y\in\mathbb{Q}_{>0}$$ a) Demonstrate that there are no injective functions with this property.b) Do exist surjective functions having this property?

$\sin\frac{\pi }{4n}\ge \frac{\sqrt 2 }{2n} , \forall n\in\mathbb{N}$

Find the number of functions $A\stackrel{f}{\longrightarrow } A$ for which there exist two functions $A\stackrel{g}{\longrightarrow } B\stackrel{h}{\longrightarrow } A$ having the properties that $g\circ h =\text{id.}$ and $h\circ g=f,$ where $B$ and $A$ are two finite sets.

Let be a surjective function $f:\mathbb{R}\longrightarrow\mathbb{R}$ that has the property that if the sequence $\left( f\left( x_n \right) \right)_{n\ge 1}$ is convergent, then the sequence $\left( x_n \right)_{n\ge 1}$ is convergent. Prove that it is continuous.

Let be two natural numbers $n\ge 2, k,$ and $k n\times n$ symmetric real matrices $A_1,A_2,\ldots ,A_k.$ Then, the following relations are equivalent: $1) \left| \sum_{i=1}^k A_i^2 \right| =0$ $2) \left| \sum_{i=1}^k A_iB_i \right| =0, \forall B_1,B_2,\ldots ,B_k\in \mathcal{M}_n\left( \mathbb{R} \right)$ $||$ denotes the determinant.

Let be a natural number $n\ge 2$ and two $n\times n$ complex matrices $A,B$ that satisfy $(AB)^3=O_n.$ Does this imply that $(BA)^3=O_n ?$

Let be a function $f$ of class $\mathcal{C}^1[a,b]$ whose derivative is positive. Prove that there exists a real number $c\in (a,b)$ such that $$f(f(b))-f(f(a))=(f'(c))^2(b-a) .$$

a) Let be a continuous function $f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} .$ Show that there exists a natural number $n_0$ and a sequence of positive real numbers $\left( x_n \right)_{n>n_0}$ that satisfy the following relation. $$n\int_0^{x_n} f(t)dt=1, n_0<\forall n\in\mathbb{N}$$ b) Prove that the sequence $\left( nx_n \right)_{n> n_0}$ is convergent and find its limit.

Let be a natural number $n$ and $2n$ real numbers $b_1,b_2,\ldots ,b_n,a_1<a_2<\cdots <a_n.$ Show thata) if $b_1,b_2,\ldots ,b_n>0,$ then there exists a polynomial $f\in\mathbb{R}[X]$ irreducible in $\mathbb{R}[X]$ such that $$f\left( a_i \right) =b_i, \forall i\in\{ 1,2,\ldots ,n \} .$$ b) there exists a polynom $g\in\mathbb{R} [X]$ of degree at least $1$ which has only real roots and such that $$g\left( a_i \right) =b_i, \forall i\in\{ 1,2,\ldots ,n \} .$$

Let $G$ be a finite group with the following property:

If $f$ is an automorphism of $G$ , then there exists $m\in\mathbb{N^\star}$ , so that $f(x)=x^{m}$ for all $x\in G$ .

Prove that G is commutative.

Marian Andronache

A function $f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}$ has the property that $\lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1.$ a) Give an example of what $f$ could be if it's continuous and $f/\text{id.}$ doesn't have a limit at $\infty .$ b) Prove that if $f$ is nondecreasing then $f/\text{id.}$ has a limit at $\infty ,$ and determine it.