RO-NMO 2009

On the sides $AB,AC$ of a triangle $ABC,$ consider the points $M,$ respectively, $N$ such that $M\neq A\neq N$ and $\frac{MB}{MA}\neq\frac{NC}{NA}.$ Show that the line $MN$ passes through a point not dependent on $M$ and $N.$

Show that for any four positive real numbers $a,b,c,d$ and four negative real numbers $e,f,g,h,$ the terms $ae+bc,ef+cg,fd+gh,da+hb$ are not all positive.

Let be a natural number $n,$ a permutation $\sigma$ of order $n,$ and $n$ nonnegative real numbers $a_1,a_2,\ldots , a_n.$ Prove the following inequality. $$\left( a_1^2+a_{\sigma (1)} \right)\left( a_2^2+a_{\sigma (2)} \right)\cdots \left( a_n^2+a_{\sigma (n)} \right)\ge \left( a_1^2+a_1 \right)\left( a_2^2+a_{2} \right)\cdots \left( a_n^2+a_n \right)$$

Let be two natural numbers $m,n\ge 2,$ two increasing finite sequences of real numbers $\left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} ,$ and the set $$\left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} .$$ Show that the set above has $n+m-1$ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.

a) Show that two real numbers $x,y>1$ chosen so that $x^y=y^x,$ are equal or there exists a positive real number $m\neq 1$ such that $x=m^{\frac{1}{m-1}}$ and $y=m^{\frac{m}{m-1}} .$ b) Solve in $\left( 1,\infty \right)^2$ the equation: $x^y+x^{x^{y-1}}=y^x+y^{y^{x-1}} .$

Let be a real number $a\in \left[ 2+\sqrt 2,4 \right] .$ Find $\inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| .$

Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ that verify the relation $$f\left( x^3+y^3 \right) =xf\left( y^2 \right) + yf\left( x^2 \right) ,$$ for all real numbers $x,y.$

We say that a natural number $n\ge 4$ is unusual if, for any $n\times n$ array of real numbers, the sum of the numbers from any $3\times 3$ compact subarray is negative, and the sum of the numbers from any $4\times 4$ compact subarray is positive. Find all unusual numbers.

Find all functions $f\in\mathcal{C}^1 [0,1]$ that satisfy $f(1)=-1/6$ and $$\int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx.$$

Find the natural numbers $n\ge 2$ which have the property that the ring of integers modulo $n$ has exactly an element that is not a sum of two squares.

Find all functions $f:[0,1]\longrightarrow [0,1]$ that are bijective, continuous and have the property that, for any continuous function $g:[0,1]\longrightarrow\mathbb{R} ,$ the following equality holds. $$\int_0^1 g\left( f(x) \right) dx =\int_0^1 g(x) dx$$