RO-NMO 2014

Let $a$ be an odd natural that is not a perfect square, and $m,n\in\mathbb{N} .$ Then

a) $\left\{ m\left( a+\sqrt a \right) \right\}\neq\left\{ n\left( a-\sqrt a \right) \right\}$ b) $\left[ m\left( a+\sqrt a \right) \right]\neq\left[ n\left( a-\sqrt a \right) \right]$

Here, $\{\},[]$ denotes the fractionary, respectively the integer part.

Let $P,Q$ be the midpoints of the diagonals $BD,$ respectively, $AC,$ of the quadrilateral $ABCD,$ and points $M,N,R,S$ on the segments $BC,CD,PQ,$ respectively $AC,$ except their extremities, such that $$\frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} .$$ Show that the center of mass of the triangle $AMN$ is situated on the segment $RS.$

Let $ABCD$ be a quadrilateral inscribed in a circle of diameter $AC.$ Fix points $E,F$ of segments $CD,$ respectively, $BC$ such that $AE$ is perpendicular to $DF$ and $AF$ is perpendicular to $BE.$ Show that $AB=AD.$

Let be a natural number $n.$ Calculate $$\sum_{k=1}^{n^2}\#\left\{ d\in\mathbb{N}| 1\le d\le k\le d^2\le n^2\wedge k\equiv 0\pmod d \right\} .$$ Here, $\#$ means cardinal.

Let be a function $f:\mathbb{N}\longrightarrow\mathbb{N}$ satisfying $\text{(i)} f(1)=1$ $\text{(ii)} f(p)=1+f(p-1),$ for any prime $p$ $\text{(iii)} f(p_1p_2\cdots p_u)=f(p_1)+f(p_2)+\cdots f(p_u),$ for any natural number $u$ and any primes $p_1,p_2,\ldots ,p_u.$ Show that $2^{f(n)}\le n^3\le 3^{f(n)},$ for any natural $n\ge 2.$

Let $n$ be a natural number, and $A$ the set of the first $n$ natural numbers. Find the number of nondecreasing functions $f:A\longrightarrow A$ that have the property $$x,y\in A\implies |f(x)-f(y)|\le |x-y|.$$

Find all continuous functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ that satisfy: $\text{(i)}\text{id}+f$ is nondecreasing $\text{(ii)}$ There is a natural number $m$ such that $\text{id}+f+f^2\cdots +f^m$ is nonincreasing.

Here, $\text{id}$ represents the identity function, and ^ denotes functional power.

Find all derivable functions that have real domain and codomain, and are equal to their second functional power.

Let $A\in\mathcal{M}_4\left(\mathbb{R}\right)$ be an invertible matrix whose trace is equal to the trace of its adjugate, which is nonzero. Show that $A^2+I$ is singular if and only if there exists a nonzero matrix in $\mathcal{M}_4\left( \mathbb{R} \right)$ that anti-commutes with it.

For a ring $A,$ and an element $a$ of it, define $s_a,d_a:A\longrightarrow A, s_a(x)=ax,d_a=xa.$ a) Prove that if $A$ is finite, then $s_a$ is injective if and only if $d_a$ is injective.b) Give example of a ring which has an element $b$ for which $s_b$ is injective and $d_b$ is not, or, conversely, $s_b$ is not injective, but $d_b$ is.

Let $I,J$ be two intervals, $\varphi :J\longrightarrow\mathbb{R}$ be a continuous function whose image doesn't contain $0,$ and $f,g:I\longrightarrow J$ be two differentiable functions such that $f'=\varphi\circ f,g'=\varphi\circ g$ and such that the image of $f-g$ contains $0.$ Show that $f$ and $g$ are the same function.

Let $f:[1,\infty )\longrightarrow (0,\infty )$ be a continuous function satisfying the following properties: $\text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}}$ $\text{(ii)}\exists\lim_{x\to\infty } \frac{1}{x}\int_1^x f(t)dt\in\mathbb{R}.$ a) Show that $\lim_{x\to\infty } \frac{f(x)}{x}=0.$ b) Prove that $\lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0.$

Let be a finite group $G$ that has an element $a\neq 1$ for which exists a prime number $p$ such that $x^{1+p}=a^{-1}xa,$ for all $x\in G.$ a) Prove that the order of $G$ is a power of $p.$ b) Show that $H:=\{x\in G|\text{ord} (x)=p\}\le G$ and $\text{ord}^2(H)>\text{ord}(G).$