RO-NMO 2010

Let $S$ be a subset with $673$ elements of the set $\{1,2,\ldots ,2010\}$ . Prove that one can find two distinct elements of $S$ , say $a$ and $b$ , such that $6$ divides $a+b$ .

Let $ABCD$ be a rectangle of centre $O$ , such that $\angle DAC=60^{\circ}$ . The angle bisector of $\angle DAC$ meets $DC$ at $S$ . Lines $OS$ and $AD$ meet at $L$ , and lines $BL$ and $AC$ meet at $M$ . Prove that lines $SM$ and $CL$ are parallel.

Each of the small squares of a $50\times 50$ table is coloured in red or blue. Initially all squares are red. A step means changing the colour of all squares on a row or on a column. a) Prove that there exists no sequence of steps, such that at the end there are exactly $2011$ blue squares. b) Describe a sequence of steps, such that at the end exactly $2010$ squares are blue.

Adriana & Lucian Dragomir

In the isosceles triangle $ABC$ , with $AB=AC$ , the angle bisector of $\angle B$ meets the side $AC$ at $B'$ . Suppose that $BB'+B'A=BC$ . Find the angles of the triangle $ABC$ .

Dan Nedeianu

Let $a,b,c$ be integers larger than $1$ . Prove that $$a(a-1)+b(b-1)+c(c-1)\le (a+b+c-4)(a+b+c-5)+4.$$

How many four digit numbers $\overline{abcd}$ simultaneously satisfy the equalities $a+b=c+d$ and $a^2+b^2=c^2+d^2$ ?

Let $VABCD$ be a regular pyramid, having the square base $ABCD$ . Suppose that on the line $AC$ lies a point $M$ such that $VM=MB$ and $(VMB)\perp (VAB)$ . Prove that $4AM=3AC$ .

Mircea Fianu

Let $a,b,c,d$ be positive integers, and let $p=a+b+c+d$ . Prove that if $p$ is a prime, then $p$ is not a divisor of $ab-cd$ .

Marian Andronache

In a triangle $ABC$ denote by $D,E,F$ the points where the angle bisectors of $\angle CAB,\angle ABC,\angle BCA$ respectively meet it's circumcircle. a) Prove that the orthocenter of triangle $DEF$ coincides with the incentre of triangle $ABC$ . b) Prove that if $\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}=0$ , then the triangle $ABC$ is equilateral.

Marin Ionescu

Prove that there is a similarity between a triangle $ABC$ and the triangle having as sides the medians of the triangle $ABC$ if and only if the squares of the lengths of the sides of triangle $ABC$ form an arithmetic sequence.

Marian Teler & Marin Ionescu

For any integer $n\ge 2$ denote by $A_n$ the set of solutions of the equation $$x=\left\lfloor\frac{x}{2}\right\rfloor+\left\lfloor\frac{x}{3}\right\rfloor+\cdots+\left\lfloor\frac{x}{n}\right\rfloor .$$ a) Determine the set $A_2\cup A_3$ . b) Prove that the set $A=\bigcup_{n\ge 2}A_n$ is finite and find $\max A$ .

Dan Nedeianu & Mihai Baluna

Consider the set $\mathcal{F}$ of functions $f:\mathbb{N}\to\mathbb{N}$ (where $\mathbb{N}$ is the set of non-negative integers) having the property that $$f(a^2-b^2)=f(a)^2-f(b)^2,\ \text{for all }a,b\in\mathbb{N},\ a\ge b.$$ a) Determine the set $\{f(1)\mid f\in\mathcal{F}\}$ . b) Prove that $\mathcal{F}$ has exactly two elements.

Nelu Chichirim

Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that $$\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.$$ Prove that $(a_n)_{n\ge0}$ is a geometric sequence.

Lucian Dragomir

Consider $v,w$ two distinct non-zero complex numbers. Prove that $$|zw+\bar{w}|\le |zv+\bar{v}|,$$ for any $z\in\mathbb{C},|z|=1$ , if and only if there exists $k\in [-1,1]$ such that $w=kv$ .

Dan Marinescu

In the plane are given $100$ points, such that no three of them are on the same line. The points are arranged in $10$ groups, any group containing at least $3$ points. Any two points in the same group are joined by a segment. a) Determine which of the possible arrangements in $10$ such groups is the one giving the minimal numbers of triangles. b) Prove that there exists an arrangement in such groups where each segment can be coloured with one of three given colours and no triangle has all edges of the same colour.

Vasile Pop

On the exterior of a non-equilateral triangle $ABC$ consider the similar triangles $ABM,BCN$ and $CAP$ , such that the triangle $MNP$ is equilateral. Find the angles of the triangles $ABM,BCN$ and $CAP$ .

Nicolae Bourbacut

Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic function and $F:\mathbb{R}\to\mathbb{R}$ given by $$F(x)=\int_0^xf(t)\ \text{d}t.$$ Prove that if $F$ has a finite derivative, then $f$ is continuous.

Dorin Andrica & Mihai Piticari

We say that a ring $A$ has property $(P)$ if any non-zero element can be written uniquely as the sum of an invertible element and a non-invertible element. a) If in $A$ , $1+1=0$ , prove that $A$ has property $(P)$ if and only if $A$ is a field. b) Give an example of a ring that is not a field, containing at least two elements, and having property $(P)$ .

Dan Schwarz

Let $G$ be a finite group of order $n$ . Define the set $$H=\{x:x\in G\text{ and }x^2=e\},$$ where $e$ is the neutral element of $G$ . Let $p=|H|$ be the cardinality of $H$ . Prove that a) $|H\cap xH|\ge 2p-n$ , for any $x\in G$ , where $xH=\{xh:h\in H\}$ . b) If $p>\frac{3n}{4}$ , then $G$ is commutative. c) If $\frac{n}{2}<p\le\frac{3n}{4}$ , then $G$ is non-commutative.

Marian Andronache

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$ , and $$I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].$$ Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$ , for any $h\in [0,1]$ . b) the sequence $(a_n)_{n\ge 1}$ , defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$ , is convergent if and only if $f(0)=0$ .

Calin Popescu