RO-NMO 2011

Let be a natural number $n$ and $n$ real numbers $a_1,a_2,\ldots ,a_n$ such that $$a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} , \forall m\in\{ 1,2,\ldots ,n \} .$$ Prove that $a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} .$

Prove that any natural number smaller or equal than the factorial of a natural number $n$ is the sum of at most $n$ distinct divisors of the factorial of $n.$

Let $ABC$ be a triangle, $I_a$ be center of the $A\text{-excircle}.$ This excircle intersects the lines $AB, AC,$ at $P,$ respectively, $Q.$ The line $PQ$ intersects the lines $I_aB,I_aC$ at $D,$ respectively, $E.$ Let $A_1$ be the intersection of $DC$ with $BE,$ and define, analogously, $B_1,C_1.$ Show that $AA_1,BB_1,CC_1$ are concurrent.

Let be a natural number $n.$ Prove that there exists a number $k\in\{ 0,1,2,\ldots n \}$ such that the floor of $2^{n+k}\sqrt 2$ is even.

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ a function having the property that $$\left| f(x+y)+\sin x+\sin y \right|\le 2,$$ for all real numbers $x,y.$ a) Prove that $\left| f(x) \right|\le 1+\cos x,$ for all real numbers $x.$ b) Give an example of what $f$ may be, if the interval $\left( -\pi ,\pi \right)$ is included in its support.

Find all numbers $n$ for which there exist three (not necessarily distinct) roots of unity of order $n$ whose sum is $1.$

Let be three positive real numbers $a,b,c.$ Show that the function $f:\mathbb{R}\longrightarrow\mathbb{R} ,$ $$f(x)=\frac{a^x}{b^x+c^x} +\frac{b^x}{a^x+c^x} +\frac{c^x}{a^x+b^x} ,$$ is nondecresing on the interval $\left[ 0,\infty \right)$ and nonincreasing on the interval $\left( -\infty ,0 \right] .$

a) Show that there exists exactly a sequence $\left( x_n,y_n \right)_{n\ge 0}$ of pairs of nonnegative integers, that satisfy the property that $\left( 1+\sqrt 33 \right)^n=x_n+y_n\sqrt 33,$ for all nonegative integers $n.$ b) Having in mind the sequence from a), prove that, for any natural prime $p,$ at least one of the numbers $y_{p-1} ,y_p$ and $y_{p+1}$ are divisible by $p.$

Prove that a ring that has a prime characteristic admits nonzero nilpotent elements if and only if its characteristic divides the number of its units.

Let be a continuous function $f:[0,1]\longrightarrow\left( 0,\infty \right)$ having the property that, for any natural number $n\ge 2,$ there exist $n-1$ real numbers $0<t_1<t_2<\cdots <t_{n-1}<1,$ such that $$\int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt.$$ Calculate $\lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} .$

The equation $x^{n+1} +x=0$ admits $0$ and $1$ as its unique solutions in a ring of order $n\ge 2.$ Prove that this ring is a skew field.

Let $f,F:\mathbb{R}\longrightarrow\mathbb{R}$ be two functions such that $f$ is nondecreasing, $F$ admits finite lateral derivates in every point of its domain, $$\lim_{x\to y^-} f(x)\le\lim_{x\to y^-}\frac{F(x)-F\left( y \right)}{x-y} ,\lim_{x\to y^+} f(x)\ge\lim_{x\to y^+}\frac{F(x)-F\left( y \right)}{x-y} ,$$ for all real numbers $y,$ and $F(0)=0.$ Prove that $F(x)=\int_0^x f(t)dt,$ for all real numbers $x.$