RO-NMO 2016

Let be a natural number $n\ge 2$ and $n$ positive real numbers $a_1,a_n,\ldots ,a_n$ that satisfy the inequalities $$\sum_{j=1}^i a_j\le a_{i+1} , \forall i\in\{ 1,2,\ldots ,n-1 \} .$$ Prove that $$\sum_{k=1}^{n-1} \frac{a_k}{a_{k+1}}\le n/2 .$$

We say that a rational number is spheric if it is the sum of three squares of rational numbers (not necessarily distinct). Prove that:

a) $7$ is not spheric. b) a rational spheric number raised to the power of any natural number greater than $1$ is spheric.

Let be a natural number $n\ge 2$ and $n$ positive real numbers $a_1,a_2,\ldots ,a_n$ whose product is $1.$ Prove that the function $f:\mathbb{R}_{>0}\longrightarrow\mathbb{R} , f(x)=\prod_{i=1}^n \left( 1+a_i^x \right) ,$ is nondecreasing.

Let be a function $f:\mathbb{R}\longrightarrow\mathbb{R}$ satisfying the conditions: $$\left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. ,$$ for all real numbers $x,y,t$ with $t\in [0,1] .$ Prove that:a) $f(b)+f(c)\le f(a)+f(d) ,$ for any real numbers $a,b,c,d$ such that $a\le b\le c\le d$ and $d-c=b-a.$ b) for any natural number $n\ge 3$ and any $n$ real numbers $x_1,x_2,\ldots ,x_n,$ the following inequality holds. $$f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right)$$

a) Let be two nonzero complex numbers $a,b.$ Show that the area of the triangle formed by the representations of the affixes $0,a,b$ in the complex plane is $\frac{1}{4}\left| \overline{a} b-a\overline{b} \right| .$ b) Let be an equilateral triangle $ABC,$ its circumcircle $\mathcal{C} ,$ its circumcenter $O,$ and two distinct points $P_1,P_2$ in the interior of $\mathcal{C} .$ Prove that we can form two triangles with sides $P_1A,P_1B,P_1C,$ respectively, $P_2A,P_2B,P_2C,$ whose areas are equal if and only if $OP_1=OP_2.$

In order to study a certain ancient language, some researchers formatted its discovered words into expressions formed by concatenating letters from an alphabet containing only two letters. Along the study, they noticed that any two distinct words whose formatted expressions have an equal number of letters, greater than $2,$ differ by at least three letters. Prove that if their observation holds indeed, then the number of formatted expressions that have $n\ge 3$ letters is at most $\left[ \frac{2^n}{n+1} \right] .$

Let be a $2\times 2$ real matrix $A$ that has the property that $\left| A^d-I_2 \right| =\left| A^d+I_2 \right| ,$ for all $d\in\{ 2014,2016 \} .$ Prove that $\left| A^n-I_2 \right| =\left| A^n+I_2 \right| ,$ for any natural number $n.$

Consider a natural number, $n\ge 2,$ and three $n\times n$ complex matrices $A,B,C$ such that $A$ is invertible, $B$ is formed by replacing the first line of $A$ with zeroes, and $C$ is formed by putting the last $n-1$ lines of $A$ above a line of zeroes. Prove that:a) $\text{rank} \left( A^{-1} B \right) = \text{rank} \left( \left( A^{-1} B\right)^2 \right) =\cdots =\text{rank} \left( \left( A^{-1} B\right)^n \right)$ b) $\text{rank} \left( A^{-1} C \right) > \text{rank} \left( \left( A^{-1} C\right)^2 \right) >\cdots >\text{rank} \left( \left( A^{-1} C\right)^n \right)$

Let be a real number $a,$ and a function $f:\mathbb{R}_{>0 }\longrightarrow\mathbb{R}_{>0 } .$ Show that the following relations are equivalent. $\text{(i)} \varepsilon\in\mathbb{R}_{>0 } \implies\left( \lim_{x\to\infty } \frac{f(x)}{x^{a+\varepsilon }} =0\wedge \lim_{x\to\infty } \frac{f(x)}{x^{a-\varepsilon }} =\infty \right)$ $\text{(ii)} \lim_{x\to\infty } \frac{\ln f(x)}{\ln x } =a$

Prove that there exists an unique sequence $\left( c_n \right)_{n\ge 1}$ of real numbers from the interval $(0,1)$ such that $$\int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } ,$$ for all natural numbers $m,$ and calculate $\lim_{k\to\infty } kc_k^k.$ Radu Pop

Let be a ring $A,$ having the property that any element in the set $A\setminus U(A)$ is nilpotent. Show that

a) $axa=0, \forall x\in A, \forall a\in A\setminus U(A) .$ b) $\mathbb{N}\ni |A|\ge 2 \implies\left(\exists d\in A\setminus \left( U(A)\cup\{ 0 \}\right) d'\in A\setminus U(A)\implies dd'=d'd=0\right) .$

Ioan Băetu

Let be a real number $a,$ and a nondecreasing function $f:\mathbb{R}\longrightarrow\mathbb{R} .$ Prove that $f$ is continuous in $a$ if and only if there exists a sequence $\left( a_n \right)_{n\ge 1}$ of real positive numbers such that $$\int_a^{a+a_n} f(x)dx+\int_a^{a-a_n} f(x)dx\le\frac{a_n}{n} ,$$ for all natural numbers $n.$ Dan Marinescu

Find the number of polynomials having nonzero pairwise distinct coefficients from a certain field that has at least three elements, and having a number of roots in this field greater or equal than the degree of these polynomials, which is equal to the order of this field minus $2.$

Marian Andronache