RO-NMO 2015

Show that among the square roots of the first $2015$ natural numbers, we cannot choose an arithmetic sequence composed of $45$ elements.

A quadratic function has the property that for any interval of length $1,$ the length of its image is at least $1.$ Show that for any interval of length $2,$ the length of its image is at least $4.$

Let be a point $P$ in the interior of a triangle $ABC.$ The lines $AP,BP,CP$ meet $BC,AC,$ respectively, $AB$ at $A_1,B_1,$ respectively, $C_1.$ If $$\mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} ,$$ show that $P$ lies on a median of $ABC.$ $\mathcal{A}$ denotes area.

Let $a,b,c,d \ge 0$ real numbers so that $a+b+c+d=1$ .Prove that $\sqrt{a+\frac{(b-c)^2}{6}+\frac{(c-d)^2}{6}+\frac{(d-b)^2}{6}} +\sqrt{b}+\sqrt{c}+\sqrt{d} \le 2.$

Find all triplets $(a,b,c)$ of nonzero complex numbers having the same absolute value and which verify the equality: $$\frac{a}{b} +\frac{b}{c}+\frac{c}{a} =-1$$

Consider a natural number $n$ for which it exist a natural number $k$ and $k$ distinct primes so that $n=p_1\cdot p_2\cdots p_k.$ a) Find the number of functions $f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\}$ that have the property that $f(1)\cdot f(2)\cdots f\left( n \right)$ divides $n.$ b) If $n=6,$ find the number of functions $f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\}$ that have the property that $f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6)$ divides $36.$

Find all functions $f,g:\mathbb{Q}\longrightarrow\mathbb{Q}$ that verify the relations $$\left\{\begin{matrix} f(g(x)+g(y))=f(g(x))+y g(f(x)+f(y))=g(f(x))+y\end{matrix}\right. ,$$ for all $x,y\in\mathbb{Q} .$

Let be a finite set $A$ of real numbers, and define the sets $S_{\pm }=\{ x\pm y| x,y\in A \} .$ Show that $\left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 .$

Find all differentiable functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ that verify the conditions: $\text{(i)} \forall x\in\mathbb{Z} f'(x) =0$ $\text{(ii)} \forall x\in\mathbb{R} f'(x)=0\implies f(x)=0$

Let be a $5\times 5$ complex matrix $A$ whose trace is $0,$ and such that $I_5-A$ is invertible. Prove that $A^5\neq I_5.$

Let be two nonnegative real numbers $a,b$ with $b>a,$ and a sequence $\left( x_n \right)_{n\ge 1}$ of real numbers such that the sequence $\left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1}$ is bounded.

Show that the sequence $\left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1}$ is convergent.

Let be three natural numbers $k,m,n$ an $m\times n$ matrix $A,$ an $n\times m$ matrix $B,$ and $k$ complex numbers $a_0,a_1,\ldots ,a_k$ such that the following conditions hold. $\text{(i)} m\ge n\ge 2$ $\text{(ii)} a_0I_m+a_1AB+a_2(AB)^2+\cdots +a_k(AB)^k=O_m$ $\text{(iii)} a_0I_m+a_1BA+a_2(BA)^2+\cdots +a_k(BA)^k\neq O_n$ Prove that $a_0=0.$

Let be a ring that has the property that all its elements are the product of two idempotent elements of it. Show that:a) $1$ is the only unit of this ring.b) this ring is Boolean.

Show that the set of all elements minus $0$ of a finite division ring that has at least $4$ elements can be partitioned into two nonempty sets $A,B$ having the property that $$\sum_{x\in A} x=\prod_{y\in B} y.$$

Let be the set $\mathcal{C} =\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f''\bigg|_{[0,1]} \right. \exists x_1,x_2\in [0,1] x_1\neq x_2\wedge \left( f\left( x_1 \right) = f\left( x_2 \right) =0\vee f\left( x_1 \right) = f'\left( x_1 \right) = 0\right) \wedge f''<1 \right\} ,$

and $f^*\in\mathcal{C}$ such that $\int_0^1\left| f^*(x) \right| dx =\sup_{f\in\mathcal{C}} \int_0^1\left| f(x) \right| dx .$

Find $\int_0^1\left| f^*(x) \right| dx$ and describe $f^*.$

Find all non-constant polynoms $f\in\mathbb{Q} [X]$ that don't have any real roots in the interval $[0,1]$ and for which there exists a function $\xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right)$ such that $h_x(x)\neq 0$ and $\int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} ,$ for all $x\in [0,1] .$