RO-NMO 2000

Let be two natural primes $1\le q \le p.$ Prove that $\left( \sqrt{p^2+q} +p\right)^2$ is irrational and its fractional part surpasses $3/4.$

Let $A,B$ be two points in a plane and let two numbers $a,b\in (0,1) .$ For each point $M$ that is not on the line $AB$ consider $P$ on the segment $AM$ and $N$ on $BM$ (both excluding the extremities) such that $BN=b\cdot BM$ and $AP=a\cdot AM.$ Find the locus of the points $M$ for which $AN=BP.$

Let be a natural number $n\ge 2$ and an expression of $n$ variables $$E\left( x_1,x_2,...,x_n\right) =x_1^2+x_2^2+\cdots +x_n^2-x_1x_2-x_2x_3-\cdots -x_{n-1}x_n -x_nx_1.$$ Determine $\sup_{x_1,...,x_n\in [0,1]} E\left( x_1,x_2,...,x_n\right)$ and the specific values at which this supremum is attained.

Let $I$ be the center of the incircle of a triangle $ABC.$ Shw that, if for any point $M$ on the segment $AB$ (extremities excluded) there exist two points $N,P$ on $BC,$ respectively, $AC$ (both excluding the extremities) such that the center of mass of $MNP$ coincides with $I,$ then $ABC$ is equilateral.

Let $\left( x_n\right)_{n\ge 1}$ be a sequence having $x_1=3$ and defined as $x_{n+1} =\left\lfloor \sqrt 2x_n\right\rfloor ,$ for every natural number $n.$ Find all values $m$ for which the terms $x_m,x_{m+1},x_{m+2}$ are in arithmetic progression, where $\lfloor\rfloor$ denotes the integer part.

Demonstrate that if $z_1,z_2\in\mathbb{C}^*$ satisfy the relation: $$z_1\cdot 2^{\big| z_1\big|} +z_2\cdot 2^{\big| z_2\big|} =\left( z_1+z_2\right)\cdot 2^{\big| z_1 +z_2\big|} ,$$ then $z_1^6=z_2^6$

Let be a tetahedron $ABCD,$ and $E$ be the projection of $D$ on the plane formed by $ABC.$ If $\mathcal{A}_{\mathcal{R}}$ denotes the area of the region $\mathcal{R},$ show that the following affirmations are equivalent:

a) $C=E\vee CE\parallel AB$ b) $M\in\overline{CD}\implies\mathcal{A}_{ABM}^2=\frac{CM^2}{CD^2}\cdot\mathcal{A}_{ABD}^2 +\left( 1-\frac{CM^2}{CD^2}\right)\cdot\mathcal{A}_{ABC}^2$

Let $f$ be a polynom of degree $3$ and having rational coefficients. Prove that, if there exist two distinct nonzero rational numbers $a,b$ and two roots $x,y$ of $f$ such that $ax+by$ is rational, then all roots of $f$ are rational.

Study the convergence of a sequence $\left( x_n\right)_{n\ge 0}$ for which $x_0\in\mathbb{R}\setminus\mathbb{Q} ,$ and $x_{n+1}\in \left\{ \frac{x_n+1}{x_n} , \frac{x_n+2}{2x_n-1}\right\} ,$ for all $n\ge 1.$

A function $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ is olympic if, any finite number of pairwise distinct elements of $\mathbb{R}^2$ at which the function takes the same value represent in the plane the vertices of a convex polygon.

Prove that if $p$ if a complex polynom of degree at least $1,$ then the function $\mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)|$ is olympic if and only if the roots of $p$ are all equal.

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ be a function that satisfies the conditions: $\text{(i)} \lim_{x\to\infty} (f\circ f) (x) =\infty =-\lim_{x\to -\infty} (f\circ f) (x)$ $\text{(ii)} f$ has Darboux's property

a) Prove that the limits of $f$ at $\pm\infty$ exist. b) Is possible for the limits from a) to be finite?

For any partition $P$ of $[0,1]$, consider the set $$\mathcal{A}(P)=\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f'\bigg|_{[0,1]}\right.\wedge\int_0^1 |f(x)|dx =1\wedge \left( y\in P\implies f (y ) =0\right)\right\} .$$ Prove that there exists a partition $P_0$ of $[0,1]$ such that $$g\in \mathcal{A}\left( P_0\right)\implies \sup_{x\in [0,1]} \big| g'(x)\big| >4\cdot \# P.$$

Here, $\# D$ denotes the natural number $d$ such that $0=x_0<x_1<\cdots <x_d=1$ is a partition $D$ of $[0,1] .$

We say that the abelian group $G$ has property (P) if, for any commutative group $H,$ any $H'\le H$ and any momorphism $\mu':H\longrightarrow G,$ there exists a morphism $\mu :H\longrightarrow G$ such that $\mu\bigg|_{H'} =\mu' .$ Show that:

a) the group $\left( \mathbb{Q}^*,\cdot \right)$ hasn't property (P). b) the group $\left( \mathbb{Q}, +\right)$ has property (P).

Prove that a nontrivial finite ring is not a skew field if and only if the equation $x^n+y^n=z^n$ has nontrivial solutions in this ring for any natural number $n.$