Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Romania National Olympiad
Let $g:\mathbb{R}\to\mathbb{R}$ be a continuous and strictly decreasing function with $g(\mathbb{R})=(-\infty,0)$ . Prove that there are no continuous functions $f:\mathbb{R}\to\mathbb{R}$ with the property that there exists a natural number $k\ge 2$ so that : $\underbrace{f\circ f\circ\ldots\circ f}_{k\text{ times}}=g$ .
2011 Romania National Olympiad
Let $A\, ,\, B\in\mathcal{M}_2(\mathbb{C})$ so that : $A^2+B^2=2AB$ .

a) Prove that : $AB=BA$ .

b) Prove that : $\text{tr}\, (A)=\text{tr}\, (B)$ .
2011 Romania National Olympiad
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.
2011 Romania National Olympiad
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)}} . $
2011 Romania National Olympiad
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.
2011 Romania National Olympiad
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. $
2010 Romania National Olympiad
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 $a+b$.
2010 Romania National Olympiad
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.
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
Let $a,b,c$ be integers larger than $1$. Prove that

$$a(a-1)+b(b-1)+c(c-1)\le (a+b+c-4)(a+b+c-5)+4.$$
2010 Romania National Olympiad
How many four digit numbers $\overline{abcd}$ simultaneously satisfy the equalities $a+b=c+d$ and $a^2+b^2=c^2+d^2$?
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
Let $a,b\in \mathbb{R}$ such that $b>a^2$. Find all the matrices $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A^2-2aA+bI_2)=0$.
2010 Romania National Olympiad
Let $A,B,C\in \mathcal{M}_n(\mathbb{R})$ such that $ABC=O_n$ and $\text{rank}\ B=1$. Prove that $AB=O_n$ or $BC=O_n$.
2010 Romania National Olympiad
Let $f:\mathbb{R}\rightarrow [0,\infty)$. Prove that $f(x+y)\ge (y+1)f(x),\ (\forall)x\in \mathbb{R}$ if and only if the function $g:\mathbb{R}\rightarrow [0,\infty),\ g(x)=e^{-x}f(x),\ (\forall)x\in \mathbb{R}$ is increasing.
2010 Romania National Olympiad
Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit.
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2010 Romania National Olympiad
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
2009 Romania National Olympiad
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. $
2009 Romania National Olympiad
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.
2009 Romania National Olympiad
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.

$$ \left( a_1^2+a_{\sigma (1)} \right)\left( a_2^2+a_{\sigma (2)} \right)\cdots \left( a_n^2+a_{\sigma (n)} \right)\ge \left( a_1^2+a_1 \right)\left( a_2^2+a_{2} \right)\cdots \left( a_n^2+a_n \right) $$
2009 Romania National Olympiad
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.
2009 Romania National Olympiad
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}} . $
2009 Romania National Olympiad
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| . $
2009 Romania National Olympiad
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. $
2009 Romania National Olympiad
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.
2009 Romania National Olympiad
Let $(t_n)_n$ a convergent sequence of real numbers, $t_n\in (0,1),\ (\forall)n\in \mathbb{N}$ and $\lim_{n\to \infty} t_n\in (0,1)$. Define the sequences $(x_n)_n$ and $(y_n)_n$ by



$$x_{n+1}=t_nx_n+(1-t_n)y_n,\ y_{n+1}=(1-t_n)x_n+t_n y_n,\ (\forall)n\in \mathbb{N}$$



and $x_0,y_0$ are given real numbers.



a) Prove that the sequences $(x_n)_n$ and $(y_n)_n$ are convergent and have the same limit.



b) Prove that if $\lim_{n\to \infty} t_n\in \{0,1\}$, then the question is false.
2009 Romania National Olympiad
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $x\in \mathbb{R}$, the limit $\lim_{h\to 0} \left|\frac{f(x+h)-f(x)}{h}\right|$ exists and it is finite. Prove that in any real point, $f$ is differentiable or it has finite one-side derivates, of the same modul, but different signs.
2009 Romania National Olympiad
Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det B\neq 0$.



a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.

b) Is the question from a) still true if $AB\neq BA$ ?
2009 Romania National Olympiad
Let $f,g,h:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is differentiable, $g$ and $h$ are monotonic, and $f'=f+g+h$. Prove that the set of the points of discontinuity of $g$ coincides with the respective set of $h$.
2009 Romania National Olympiad
Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and

$$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$
2009 Romania National Olympiad
a) Show that the set of nilpotents of a finite, commutative ring, is closed under each of the operations of the ring.

b) Prove that the number of nilpotents of a finite, commutative ring, divides the number of divisors of zero of the ring.
2009 Romania National Olympiad
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.
2009 Romania National Olympiad
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.

$$ \int_0^1 g\left( f(x) \right) dx =\int_0^1 g(x) dx $$
2008 Romania National Olympiad
Let $ ABC$ be an acute angled triangle with $ \angle B > \angle C$. Let $ D$ be the foot of the altitude from $ A$ on $ BC$, and let $ E$ be the foot of the perpendicular from $ D$ on $ AC$. Let $ F$ be a point on the segment $ (DE)$. Show that the lines $ AF$ and $ BF$ are perpendicular if and only if $ EF\cdot DC = BD \cdot DE$.
2008 Romania National Olympiad
A rectangle can be divided by parallel lines to its sides into 200 congruent squares, and also in 288 congruent squares. Prove that the rectangle can also be divided into 392 congruent squares.
2008 Romania National Olympiad
Let $ p,q,r$ be 3 prime numbers such that $ 5\leq p <q<r$. Knowing that $ 2p^2-r^2 \geq 49$ and $ 2q^2-r^2\leq 193$, find $ p,q,r$.
2008 Romania National Olympiad
Let $ ABCD$ be a rectangle with center $ O$, $ AB\neq BC$. The perpendicular from $ O$ to $ BD$ cuts the lines $ AB$ and $ BC$ in $ E$ and $ F$ respectively. Let $ M,N$ be the midpoints of the segments $ CD,AD$ respectively. Prove that $ FM \perp EN$.
2008 Romania National Olympiad
A tetrahedron has the side lengths positive integers, such that the product of any two opposite sides equals 6. Prove that the tetrahedron is a regular triangular pyramid in which the lateral sides form an angle of at least 30 degrees with the base plane.
2008 Romania National Olympiad
a) We call admissible sequence a sequence of 4 even digits in which no digits appears more than two times. Find the number of admissible sequences.



b) For each integer $ n\geq 2$ we denote $ d_n$ the number of possibilities of completing with even digits an array with $ n$ rows and 4 columns, such that



(1) any row is an admissible sequence; (2) the sequence 2, 0, 0, 8 appears exactly ones in the array.



Find the values of $ n$ for which the number $ \frac {d_{n+1}}{d_n}$ is an integer.
2008 Romania National Olympiad
Let $ a,b \in [0,1]$. Prove that $$ \frac 1{1+a+b} \leq 1 - \frac {a+b}2 + \frac {ab}3.$$
2008 Romania National Olympiad
Let $ ABCDA'B'C'D'$ be a cube. On the sides $ (A'D')$, $ (A'B')$ and $ (A'A)$ we consider the points $ M_1$, $ N_1$ and $ P_1$ respectively. On the sides $ (CB)$, $ (CD)$ and $ (CC')$ we consider the points $ M_2$, $ N_2$ and $ P_2$ respectively. Let $ d_1$ be the distance between the lines $ M_1N_1$ and $ M_2N_2$, $ d_2$ be the distance between the lines $ N_1P_1$ and $ N_2P_2$, and $ d_3$ be the distance between the lines $ P_1M_1$ and $ P_2M_2$. Suppose that the distances $ d_1$, $ d_2$ and $ d_3$ are pairwise distinct. Prove that the lines $ M_1M_2$, $ N_1N_2$ and $ P_1P_2$ are concurrent.
2008 Romania National Olympiad
Find functions $ f: \mathbb{N} \rightarrow \mathbb{N}$, such that $ f(x^2 + f(y)) = xf(x) + y$, for $ x,y \in \mathbb{N}$.
2008 Romania National Olympiad
a) Prove that

$$ \dfrac{1}{2} + \dfrac{1}{3} + ... + \dfrac{1}{2^{2n}} > n,

$$

for all positive integers $ n$.



b) Prove that for every positive integer $ n$ we have $ \min\left\{ k \in \mathbb{Z}, k\geq 2 \mid \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{k}>n \right\} > 2^n$.
2008 Romania National Olympiad
Let $ n$ be a positive integer and let $ a_i$ be real numbers, $ i = 1,2,\ldots,n$ such that  $ |a_i|\leq 1$ and $ \sum_{i=1}^n a_i = 0$.

Show that $ \sum_{i=1}^n |x - a_i|\leq n$, for every $ x\in \mathbb{R}$ with $ |x|\le 1$.
2008 Romania National Olympiad
On the sides of triangle $ ABC$ we consider points $ C_1,C_2 \in (AB), B_1,B_2 \in (AC), A_1,A_2 \in (BC)$ such that triangles $ A_1,B_1,C_1$ and $ A_2B_2C_2$ have a common centroid.

Prove that sets $ [A_1,B_1]\cap [A_2B_2], [B_1C_1]\cap[B_2C_2], [C_1A_1]\cap [C_2A_2]$ are not empty.
2008 Romania National Olympiad
Let $ ABC$ be a triangle and the points $ D\in (BC)$, $ E\in (CA)$, $ F\in (AB)$ such that $$ \frac {BD}{DC} = \frac {CE}{EA} = \frac {AF}{FB}.$$ Prove that if the circumcenters of the triangles $ DEF$ and $ ABC$ coincide then $ ABC$ is equilateral.
2008 Romania National Olympiad
Let $ a,b,c$ be 3 complex numbers such that $$ a|bc| + b|ca| + c|ab| = 0.$$ Prove that $$ |(a-b)(b-c)(c-a)| \geq 3\sqrt 3 |abc|.$$
2008 Romania National Olympiad
Let $ A=\{1,2,\ldots, 2008\}$. We will say that set $ X$ is an $ r$-set if $ \emptyset \neq X \subset A$, and $ \sum_{x\in X} x \equiv r \pmod 3$. Let $ X_r$, $ r\in\{0,1,2\}$ be the set of $ r$-sets.



Find which one of $ X_r$ has the most elements.
2008 Romania National Olympiad
We consider the proposition $ p(n)$: $ n^2+1$ divides $ n!$, for positive integers $ n$. Prove that there are infinite values of $ n$ for which $ p(n)$ is true, and infinite values of $ n$ for which $ p(n)$ is false.
2008 Romania National Olympiad
Let $ f : (0,\infty) \to \mathbb R$ be a continous function such that the sequences $ \{f(nx)\}_{n\geq 1}$ are nondecreasing for any real number $ x$. Prove that $ f$ is nondecreasing.
2008 Romania National Olympiad
Let $ A$ be a $ n\times n$ matrix with complex elements. Prove that $ A^{-1} = \overline{A}$ if and only if there exists an invertible matrix $ B$ with complex elements such that $ A= B^{-1} \cdot \overline{B}$.
2008 Romania National Olympiad
Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that

$$ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,$$ for all $ a\neq b \in \mathbb R$.



Prove that $ f''(c)=0$.
2008 Romania National Olympiad
Let $ A=(a_{ij})_{1\leq i,j\leq n}$ be a real $ n\times n$ matrix, such that $ a_{ij} + a_{ji} = 0$, for all $ i,j$. Prove that for all non-negative real numbers $ x,y$ we have $$ \det(A+xI_n)\cdot \det(A+yI_n) \geq \det (A+\sqrt{xy}I_n)^2.$$
2008 Romania National Olympiad
Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)=0$ and for all nonnegative $ x$ we have

$$ xf(x) \geq \int^x_0 f(t) dt ,$$ then $ f$ admits primitives on $ [0,\infty)$.
2008 Romania National Olympiad
Let $ f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $ f'$ on $ [0,1]$. Prove that if $ f\left( \frac 12\right) = 0$, then

$$ \int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.$$
2008 Romania National Olympiad
Let $ A$ be a unitary finite ring with $ n$ elements, such that the equation $ x^n=1$ has a unique solution in $ A$, $ x=1$. Prove that



a) $ 0$ is the only nilpotent element of $ A$;



b) there exists an integer $ k\geq 2$, such that the equation $ x^k=x$ has $ n$ solutions in $ A$.
2008 Romania National Olympiad
Let $ \mathcal G$ be the set of all finite groups with at least two elements.



a) Prove that if $ G\in \mathcal G$, then the number of morphisms $ f: G\to G$ is at most $ \sqrt [p]{n^n}$, where $ p$ is the largest prime divisor of $ n$, and $ n$ is the number of elements in $ G$.



b) Find all the groups in $ \mathcal G$ for which the inequality at point a) is an equality.
2007 Romania National Olympiad
Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ (real $2\times 2$ matrices), that satisfy $A^{2}+B^{2}=AB$. Prove that $(AB-BA)^{2}=O_{2}$.
2007 Romania National Olympiad
Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ be two points in the image of $f$ (that is, there exists $x,y$ such that $f(x)=a$ and $f(y)=b$).



Show that there is an interval $I$ such that $f(I)=[a,b]$.
2007 Romania National Olympiad
Let $n\geq 2$ be an integer and denote by $H_{n}$ the set of column vectors $^{T}(x_{1},\ x_{2},\ \ldots, x_{n})\in\mathbb{R}^{n}$, such that $\sum |x_{i}|=1$.



Prove that there exist only a finite number of matrices $A\in\mathcal{M}_{n}(\mathbb{R})$ such that the linear map $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ given by $f(x)=Ax$ has the property $f(H_{n})=H_{n}$.



CommentIn the contest, the problem was given with a) and b):



a) Prove the above for $n=2$;



b) Prove the above for $n\geq 3$ as well.
2007 Romania National Olympiad
Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$.



a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$.



b) Give an example of a non-constant function $f$ with property $(P)$.



c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.
2007 Romania National Olympiad
Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$, $f(1)=1$. Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$) and find all functions $f\in\mathcal{F}$ for which this minimum is attained.



CommentIn the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''.
2007 Romania National Olympiad
Let $f: [0,1]\rightarrow(0,+\infty)$ be a continuous function.



a) Show that for any integer $n\geq 1$, there is a unique division $0=a_{0}<a_{1}<\ldots<a_{n}=1$ such that $\int_{a_{k}}^{a_{k+1}}f(x)\, dx=\frac{1}{n}\int_{0}^{1}f(x)\, dx$ holds for all $k=0,1,\ldots,n-1$.



b) For each $n$, consider the $a_{i}$ above (that depend on $n$) and define $b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$. Show that the sequence $(b_{n})$ is convergent and compute it's limit.
2007 Romania National Olympiad
Let $n\geq 1$ be an integer. Find all rings $(A,+,\cdot)$ such that all $x\in A\setminus\{0\}$ satisfy $x^{2^{n}+1}=1$.
2007 Romania National Olympiad
Let $n\geq 3$ be an integer and $S_{n}$ the permutation group. $G$ is a subgroup of $S_{n}$, generated by $n-2$ transpositions. For all $k\in\{1,2,\ldots,n\}$, denote by $S(k)$ the set $\{\sigma(k) \ : \ \sigma\in G\}$.



Show that for any $k$, $|S(k)|\leq n-1$.
2007 Romania National Olympiad
Let $a, b, c, d \in \mathbb{N^{*}}$ such that the equation $$x^{2}-(a^{2}+b^{2}+c^{2}+d^{2}+1)x+ab+bc+cd+da=0 $$ has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.
2007 Romania National Olympiad
Let $ABC$ be an acute angled triangle and point $M$ chosen differently from $A,B,C$. Prove that $M$ is the orthocenter of triangle $ABC$ if and only if $$\frac{BC}{MA}\vec{MA}+\frac{CA}{MB}\vec{MB}+\frac{AB}{MC}\vec{MC}= \vec{0}$$
2007 Romania National Olympiad
The plane is divided into strips of width $1$ by parallel lines (a strip - the region between two parallel lines). The points from the interior of each strip are coloured with red or white, such that in each strip only one color is used (the points of a strip are coloured with the same color). The points on the lines are not coloured. Show that there is an equilateral triangle of side-length $100$, with all vertices of the same colour.