Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Romania National Olympiad
Let be a ring $ A, $ having the property that any element in the set $ A\setminus U(A) $ is nilpotent. Show that



a) $ axa=0,\quad\forall x\in A,\quad \forall a\in A\setminus U(A) . $

b) $ \mathbb{N}\ni |A|\ge 2 \implies\left(\exists d\in A\setminus \left( U(A)\cup\{ 0 \}\right) \quad d'\in A\setminus U(A)\implies dd'=d'd=0\right) . $





Ioan Băetu
2016 Romania National Olympiad
Let be a real number $ a, $ and a nondecreasing function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ Prove that $ f $ is continuous in $ a $ if and only if there exists a sequence $ \left( a_n \right)_{n\ge 1} $ of real positive numbers such that

$$ \int_a^{a+a_n} f(x)dx+\int_a^{a-a_n} f(x)dx\le\frac{a_n}{n} , $$for all natural numbers $ n. $





Dan Marinescu
2016 Romania National Olympiad
Find the number of polynomials having nonzero pairwise distinct coefficients from a certain field that has at least three elements, and having a number of roots in this field greater or equal than the degree of these polynomials, which is equal to the order of this field minus $ 2. $





Marian Andronache
2015 Romania National Olympiad
Show that among the square roots of the first $ 2015 $ natural numbers, we cannot choose an arithmetic sequence composed of $ 45 $ elements.
2015 Romania National Olympiad
A quadratic function has the property that for any interval of length $ 1, $ the length of its image is at least $ 1. $

Show that for any interval of length $ 2, $ the length of its image is at least $ 4. $
2015 Romania National Olympiad
Let be a point $ P $ in the interior of a triangle $ ABC. $ The lines $ AP,BP,CP $ meet $ BC,AC, $ respectively, $ AB $ at $ 

A_1,B_1, $ respectively, $ C_1. $ If

$$ \mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} , $$show that $ P $ lies on a median of $ ABC. $



$ \mathcal{A} $ denotes area.
2015 Romania National Olympiad
Let $a,b,c,d \ge 0$ real numbers so that $a+b+c+d=1$.Prove that

$\sqrt{a+\frac{(b-c)^2}{6}+\frac{(c-d)^2}{6}+\frac{(d-b)^2}{6}} +\sqrt{b}+\sqrt{c}+\sqrt{d} \le 2.$
2015 Romania National Olympiad
Find all triplets $ (a,b,c) $ of nonzero complex numbers having the same absolute value and which verify the equality:

$$ \frac{a}{b} +\frac{b}{c}+\frac{c}{a} =-1 $$
2015 Romania National Olympiad
Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $



a) Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n. $

b) If $ n=6, $ find the number of functions $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $ that have the property that $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $ divides $ 36. $
2015 Romania National Olympiad
Find all functions $ f,g:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify the relations

$$ \left\{\begin{matrix} f(g(x)+g(y))=f(g(x))+y \\
g(f(x)+f(y))=g(f(x))+y\end{matrix}\right. , $$for all $ x,y\in\mathbb{Q} . $
2015 Romania National Olympiad
Let be a finite set $ A $ of real numbers, and define the sets $ S_{\pm }=\{ x\pm y| x,y\in A \} . $

Show that $ \left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 . $
2015 Romania National Olympiad
Find all differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the conditions:

$ \text{(i)}\quad\forall x\in\mathbb{Z} \quad f'(x) =0 $



$ \text{(ii)}\quad\forall x\in\mathbb{R}\quad f'(x)=0\implies f(x)=0 $
2015 Romania National Olympiad
Let be a $ 5\times 5 $ complex matrix $ A $ whose trace is $ 0, $ and such that $ I_5-A $ is invertible.

Prove that $ A^5\neq I_5. $
2015 Romania National Olympiad
Let be two nonnegative real numbers $ a,b $ with $ b>a, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers such that the sequence $ \left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1} $ is bounded.



Show that the sequence $ \left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1} $ is convergent.
2015 Romania National Olympiad
Let be three natural numbers $ k,m,n $ an $ m\times n $ matrix $ A, $ an $ n\times m $ matrix $ B, $ and $ k $ complex numbers $ a_0,a_1,\ldots ,a_k $ such that the following conditions hold.



$ \text{(i)}\quad m\ge n\ge 2 $

$ \text{(ii)}\quad a_0I_m+a_1AB+a_2(AB)^2+\cdots +a_k(AB)^k=O_m $

$ \text{(iii)}\quad a_0I_m+a_1BA+a_2(BA)^2+\cdots +a_k(BA)^k\neq O_n $



Prove that $ a_0=0. $
2015 Romania National Olympiad
Let be a ring that has the property that all its elements are the product of two idempotent elements of it. Show that:

a) $ 1 $ is the only unit of this ring.

b) this ring is Boolean.
2015 Romania National Olympiad
Show that the set of all elements minus $ 0 $ of a finite division ring that has at least $ 4 $ elements can be partitioned into two nonempty sets $ A,B $ having the property that

$$ \sum_{x\in A} x=\prod_{y\in B} y. $$
2015 Romania National Olympiad
Let be the set $ \mathcal{C} =\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f''\bigg|_{[0,1]} \right.\quad\exists x_1,x_2\in [0,1]\quad x_1\neq x_2\wedge \left( f\left( 

x_1 \right) = f\left( x_2 \right) =0\vee  f\left( 

x_1 \right) = f'\left( x_1 \right) = 0\right)  \wedge f''<1 \right\} , $



and $ f^*\in\mathcal{C} $ such that $ \int_0^1\left| f^*(x) \right| dx =\sup_{f\in\mathcal{C}} \int_0^1\left| f(x) \right| dx .  $



Find $ \int_0^1\left| f^*(x) \right| dx $ and describe $ f^*. $
2015 Romania National Olympiad
Find all non-constant polynoms $ f\in\mathbb{Q} [X] $ that don't have any real roots in the interval $ [0,1] $ and for which there exists a function $ \xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right) $ such that $ h_x(x)\neq 0 $ and $ \int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} , $ for all $ x\in [0,1] . $
2014 Romania National Olympiad
Find x, y, z $\in Z$

$x^2+y^2+z^2=2^n(x+y+z)$

$n\in N$
2014 Romania National Olympiad
Let $ a $ be an odd natural that is not a perfect square, and $ m,n\in\mathbb{N} . $ Then



a) $ \left\{ m\left( a+\sqrt a \right) \right\}\neq\left\{ n\left( a-\sqrt a \right) \right\} $

b) $ \left[ m\left( a+\sqrt a \right) \right]\neq\left[ n\left( a-\sqrt a \right) \right] $



Here, $ \{\},[] $ denotes the fractionary, respectively the integer part.
2014 Romania National Olympiad
Let $ P,Q $ be the midpoints of the diagonals $ BD, $ respectively, $ AC, $ of the quadrilateral $ ABCD, $ and points $ M,N,R,S $ on the segments $ BC,CD,PQ, $ respectively $ AC, $ except their extremities, such that

$$ \frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} . $$Show that the center of mass of the triangle $ AMN $ is situated on the segment $ RS. $
2014 Romania National Olympiad
Let $ ABCD $ be a quadrilateral inscribed in a circle of diameter $ AC. $ Fix points $ E,F $ of segments $ CD, $ respectively, $ BC $ such that $ AE $ is perpendicular to $ DF $ and $ AF $ is perpendicular to $ BE. $

Show that $ AB=AD. $
2014 Romania National Olympiad
Let be a natural number $ n. $ Calculate

$$ \sum_{k=1}^{n^2}\#\left\{ d\in\mathbb{N}| 1\le d\le k\le d^2\le n^2\wedge k\equiv 0\pmod d  \right\} . $$

Here, $ \# $ means cardinal.
2014 Romania National Olympiad
Let be a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ satisfying

$ \text{(i)} f(1)=1 $

$ \text{(ii)} f(p)=1+f(p-1), $ for any prime $ p $

$ \text{(iii)} f(p_1p_2\cdots p_u)=f(p_1)+f(p_2)+\cdots f(p_u), $ for any natural number $ u $ and any primes $ p_1,p_2,\ldots ,p_u. $



Show that $ 2^{f(n)}\le n^3\le 3^{f(n)}, $ for any natural $ n\ge 2. $
2014 Romania National Olympiad
Let $ n $ be a natural number, and $ A $ the set of the first $ n $ natural numbers. Find the number of nondecreasing functions $ f:A\longrightarrow A $ that have the property

$$ x,y\in A\implies |f(x)-f(y)|\le |x-y|. $$
2014 Romania National Olympiad
Let $n \in \mathbb{N} , n \ge 2$ and $ a_0,a_1,a_2,\cdots,a_n \in \mathbb{C} ; a_n \not = 0 $. Then:



P. $|a_nz^n + a_{n-1}z^z{n-1} + \cdots + a_1z + a_0 | \le |a_n+a_0|$ for any $z \in \mathbb{C}, |z|=1$

Q. $a_1=a_2=\cdots=a_{n-1}=0$   and   $a_0/a_n \in [0,\infty)$



Prove that $ P \Longleftrightarrow Q$
2014 Romania National Olympiad
Find all continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy:

$ \text{(i)}\text{id}+f $ is nondecreasing

$ \text{(ii)} $ There is a natural number $ m $ such that $ \text{id}+f+f^2\cdots +f^m $ is nonincreasing.



Here, $ \text{id} $ represents the identity function, and  ^  denotes functional power.
2014 Romania National Olympiad
Find all derivable functions that have real domain and codomain, and are equal to their second functional power.
2014 Romania National Olympiad
Let $A,B\in M_n(C)$ be two square matrices satisfying $A^2+B^2 = 2AB$.



1.Prove that $\det(AB-BA)=0$.

2.If $rank(A-B)=1$, then prove that $AB=BA$.
2014 Romania National Olympiad
Let $ A\in\mathcal{M}_4\left(\mathbb{R}\right) $ be an invertible matrix whose trace is equal to the trace of its adjugate, which is nonzero. Show that $ A^2+I $ is singular if and only if there exists a nonzero matrix in $ \mathcal{M}_4\left( \mathbb{R} \right) $ that anti-commutes with it.
2014 Romania National Olympiad
For a ring $ A, $ and an element $ a $ of it, define $ s_a,d_a:A\longrightarrow A, s_a(x)=ax,d_a=xa.$



a) Prove that if $ A $ is finite, then $ s_a $ is injective if and only if $ d_a $ is injective.

b) Give example of a ring which has an element $ b $ for which $ s_b $ is injective and $ d_b $ is not, or, conversely, $ s_b $ is not injective, but $ d_b $ is.
2014 Romania National Olympiad
Let $ I,J $ be two intervals, $ \varphi :J\longrightarrow\mathbb{R} $ be a continuous function whose image doesn't contain $ 0, $ and $ f,g:I\longrightarrow J $ be two differentiable functions such that $ f'=\varphi\circ f,g'=\varphi\circ g $ and such that the image of $ f-g $ contains $ 0. $

Show that $ f $ and $ g $ are the same function.
2014 Romania National Olympiad
Let $ f:[1,\infty )\longrightarrow (0,\infty ) $ be a continuous function satisfying the following properties:



$ \text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}} $

$ \text{(ii)}\exists\lim_{x\to\infty } \frac{1}{x}\int_1^x f(t)dt\in\mathbb{R}. $



a) Show that $ \lim_{x\to\infty } \frac{f(x)}{x}=0. $

b) Prove that $ \lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0. $
2014 Romania National Olympiad
Let be a finite group $ G $ that has an element $ a\neq 1 $ for which exists a prime number $ p $ such that $ x^{1+p}=a^{-1}xa, $ for all $ x\in G. $



a) Prove that the order of $ G $ is a power of $ p. $

b) Show that $ H:=\{x\in G|\text{ord} (x)=p\}\le G $ and $ \text{ord}^2(H)>\text{ord}(G). $
2013 Romania National Olympiad
A series of numbers is called complete if it has non-zero natural terms and any nonzero integer has at least one among multiple series. Show that the arithmetic progression is a complete sequence if and only if it divides the first term relationship.
2013 Romania National Olympiad
Given    $f:\mathbb{R}\to \mathbb{R}$    an arbitrary function and   $g:\mathbb{R}\to \mathbb{R}$   a function of the second degree, with the property:

for any real numbers m and n equation   $f\left( x \right)=mx+n$   has solutions if and only if the equation  $g\left( x \right)=mx+n$   has solutions

Show that the functions $f$ and $g$ are equal.
2013 Romania National Olympiad
Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$

a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$

b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$
2013 Romania National Olympiad
Consider a nonzero integer number $n$ and the function $f:\mathbb{N}\to \mathbb{N}$ by

$$ f(x) = \begin{cases}

		\frac{x}{2} & \text{if } x \text{ is even} \\
		\frac{x-1}{2} + 2^{n-1} & \text{if } x \text{ is odd}

\end{cases}. 

$$ Determine the set: $$ 

	A = \{ x\in \mathbb{N} \mid \underbrace{\left( f\circ f\circ ....\circ f \right)}_{n\ f\text{'s}}\left( x \right)=x \}. 

	$$
2013 Romania National Olympiad
Solve the following equation  ${{2}^{{{\sin }^{4}}x-{{\cos }^{2}}x}}-{{2}^{{{\cos }^{4}}x-{{\sin }^{2}}x}}=\cos 2x$
2013 Romania National Olympiad
To be considered the following complex and distinct $a,b,c,d$. Prove that the following affirmations are equivalent:

i)For every $z\in \mathbb{C}$ the inequality takes place :$\left| z-a \right|+\left| z-b \right|\ge \left| z-c \right|+\left| z-d \right|$;

ii)There is $t\in \left( 0,1 \right)$ so that $c=ta+\left( 1-t \right)b$ si $d=\left( 1-t \right)a+tb$
2013 Romania National Olympiad
Find all injective functions$f:\mathbb{Z}\to \mathbb{Z}$ that satisfy:

$\left| f\left( x \right)-f\left( y \right) \right|\le \left| x-y \right|$ ,for any $x,y\in \mathbb{Z}$.
2013 Romania National Olympiad
a)Prove that $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{{{2}^{m}}}<m$, for any $m\in {{\mathbb{N}}^{*}}$.

b)Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that

$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$
2013 Romania National Olympiad
Given A, non-inverted matrices of order n with real elements, $n\ge 2$  and given ${{A}^{*}}$adjoin matrix A. Prove that $tr({{A}^{*}})\ne -1$ if and only if the matrix ${{I}_{n}}+{{A}^{*}}$  is invertible.
2013 Romania National Olympiad
Whether $m$ and $n$ natural numbers, $m,n\ge 2$. Consider matrices, ${{A}_{1}},{{A}_{2}},...,{{A}_{m}}\in {{M}_{n}}(R)$ not all nilpotent. Demonstrate that there is an integer number $k>0$ such that ${{A}^{k}}_{1}+{{A}^{k}}_{2}+.....+{{A}^{k}}_{m}\ne {{O}_{n}}$
2013 Romania National Olympiad
A function $$\text{f:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}$$ is called contract if,  for every numbers $x,y\in \text{(0,}\infty \text{)}$ we have, $\underset{n\to \infty }{\mathop{\lim }}\,\left( {{f}^{n}}\left( x \right)-{{f}^{n}}\left( y \right) \right)=0$ where ${{f}^{n}}=\underbrace{f\circ f\circ ...\circ f}_{n\ f\text{'s}}$

a) Consider $$f:\text{(0,}\infty \text{) }\to \text{(0,}\infty \text{)}$$ a function contract, continue with the property that has a fixed point, that existing ${{x}_{0}}\in \text{(0,}\infty \text{) }$ there so that $f\left( {{x}_{0}} \right)={{x}_{0}}.$ Show that $f\left( x \right)>x,$ for every $x\in \text{(0,}{{x}_{0}}\text{)}\,$ and $f\left( x \right)<x$, for every $x\in \text{(}{{x}_{0}}\text{,}\infty \text{)}\,$.

b) Show that the given function $$f\text{:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}$$ given by $f\left( x \right)=x+\frac{1}{x}$ is contracted but has no fix number.
2013 Romania National Olympiad
a) Consider$$f\text{:}\left[ \text{0,}\infty  \right)\to \left[ \text{0,}\infty  \right)$$ a differentiable and convex function .Show that $f\left( x \right)\le x$,  for every $x\ge 0$, than ${f}'\left( x \right)\le 1$ ,for every $x\ge 0$

b) Determine $$f\text{:}\left[ \text{0,}\infty  \right)\to \left[ \text{0,}\infty  \right)$$ differentiable and convex functions which have the property that $f\left( 0 \right)=0\,$, and ${f}'\left( x \right)f\left( f\left( x \right) \right)=x$, for every $x\ge 0$
2013 Romania National Olympiad
Determine continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $\left( {{a}^{2}}+ab+{{b}^{2}} \right)\int\limits_{a}^{b}{f\left( x \right)dx=3\int\limits_{a}^{b}{{{x}^{2}}f\left( x \right)dx,}}$ for every $a,b\in \mathbb{R}$ .
2013 Romania National Olympiad
Given a ring $\left( A,+,\cdot  \right)$ that meets both of the following conditions:

(1) $A$ is not a field, and

(2) For every non-invertible element $x$ of $ A$, there is an integer $m>1$ (depending on $x$) such that $x=x^2+x^3+\ldots+x^{2^m}$.

Show that

(a) $x+x=0$ for every $x \in A$, and

(b) $x^2=x$ for every non-invertible $x\in A$.
2013 Romania National Olympiad
Given $a\in (0,1)$ and $C$ the set of increasing functions

$f:[0,1]\to [0,\infty )$ such that  $\int\limits_{0}^{1}{f(x)}dx=1$ . Determine:

$(a)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{f(x)dx}$

$(b)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{{{f}^{2}}(x)dx}$
2013 Romania National Olympiad
Given $n\ge 2$ a natural number, $(K,+,\cdot )$ a body with commutative property that $\underbrace{1+...+}_{m}1\ne 0,m=2,...,n,f\in K[X]$ a polynomial of degree $n$ and $G$ a subgroup of the additive group $(K,+,\cdot )$, $G\ne K.$Show that there is $a\in K$ so$f(a)\notin G$.
2012 Romania National Olympiad
The altitude $[BH]$ dropped onto the hypotenuse of a triangle $ABC$ intersects the bisectors $[AD]$ and $[CE]$ at $Q$ and $P$ respectively. Prove that the line passing through the midpoints of the segments $[QD]$ and $[PE]$ is parallel to the line $AC$ .
2012 Romania National Olympiad
Find all functions $f:\mathbb{R}\to\mathbb{R}$ with the following property: for any open bounded interval $I$, the set $f(I)$ is an open interval having the same length with $I$ .
2012 Romania National Olympiad
Prove that if $n\ge 2$ is a natural number and $x_1,x_2,\ldots,x_n$ are positive real numbers, then:



$$4\left(\frac {x_1^3-x_2^3}{x_1+x_2}+\frac {x_2^3-x_3^3}{x_2+x_3}+\ldots+\frac {x_{n-1}^3-x_n^3}{x_{n-1}+x_n}+\frac {x_n^3-x_1^3}{x_n+x_1}\right)\le \\ \\ 

\le(x_1-x_2)^2+(x_2-x_3)^2+\ldots+(x_{n-1}-x_n)^2+(x_n-x_1)^2\, .$$
2012 Romania National Olympiad
On a table there are $k\ge 2$ piles having $n_1,n_2,\ldots,n_k$ pencils respectively. A move consists in choosing two piles having $a$ and $b$ pencils respectively, $a\ge b$ and transferring $b$ pencils from the first pile to the second one. Find the necessary and sufficient condition for $n_1,n_2,\ldots,n_k$ , such that there exists a succession of moves through which all pencils are transferred to the same pile.
2012 Romania National Olympiad
Let $M=\{x\in\mathbb{C}\, |\, |z|=1,\ \text{Re}\, z\in\mathbb{Q}\}\, .$ Prove that there exist infinitely many equilateral triangles in the complex plane having all affixes of their vertices in the set $M$ .
2012 Romania National Olympiad
Let $a$ , $b$ and $c$ be three complex numbers such that $a+b+c=0$ and $|a|=|b|=|c|=1$ . Prove that:



$$3\le |z-a|+|z-b|+|z-c|\le 4,$$

for any $z\in\mathbb{C}$ , $|z|\le 1\, .$
2012 Romania National Olympiad
Let $a,b\in\mathbb{R}$ with $0<a<b$ . Prove that:









a) $2\sqrt {ab}\le\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\le a+b$ , for $x,y,z\in [a,b]\, .$



b) $\left\{\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\, |\, x,y,z\in [a,b]\right\}=[2\sqrt {ab},a+b]\, .$
2012 Romania National Olympiad
Let $n$ and $m$ be two natural numbers, $m\ge n\ge 2$ . Find the number of injective functions



$$f\colon\{1,2,\ldots,n\}\to\{1,2,\ldots,m\}$$

such that there exists a unique number $i\in\{1,2,\ldots,n-1\}$ for which $f(i)>f(i+1)\, .$
2012 Romania National Olympiad
Let $f,g\colon [0,1]\to [0,1]$ be two functions such that $g$ is monotonic, surjective and $|f(x)-f(y)|\le |g(x)-g(y)|$ , for any $x,y\in [0,1]$ .









a) Prove that $f$ is continuous and that there exists some $x_0\in [0,1]$ with $f(x_0)=g(x_0)$ .

b) Prove that the set $\{x\in [0,1]\, |\, f(x)=g(x)\}$ is a closed interval.
2012 Romania National Olympiad
Let $n$ and $k$ be two natural numbers such that $n\ge 2$ and $1\le k\le n-1$ . Prove  that if the matrix $A\in\mathcal{M}_n(\mathbb{C})$ has exactly $k$ minors of order $n-1$ equal to $0$ , then $\det (A)\ne 0$ .
2012 Romania National Olympiad
Let $A,B\in\mathcal{M}_4(\mathbb{R})$ such that $AB=BA$ and $\det (A^2+AB+B^2)=0$ . Prove that:



$$\det (A+B)+3\det (A-B)=6\det (A)+6\det (B)\ .$$
2012 Romania National Olympiad
Find all differentiable functions $f\colon [0,\infty)\to [0,\infty)$ for which $f(0)=0$ and $f^{\prime}(x^2)=f(x)$ for any $x\in [0,\infty)$ .
2012 Romania National Olympiad
Let $f\colon [0,\infty)\to\mathbb{R}$ be a continuous function such that $\int_0^nf(x)f(n-x)\ \text{d}x=\int_0^nf^2(x)\ \text{d}x$ , for any natural number $n\ge 1$ . Prove that $f$ is a periodic function.
2012 Romania National Olympiad
 Let $(R,+,\cdot)$ be a ring and let $f$ be a surjective endomorphism of $R$ such that $[x,f(x)]=0$ for any $x\in R$ , where $[a,b]=ab-ba$ , $a,b\in R$ . Prove that:









a) $[x,f(y)]=[f(x),y]$ and $x[x,y]=f(x)[x,y]$ , for any $x,y\in R\ ;$

b) If $R$ is a division ring and $f$ is different from the identity function, then $R$ is commutative.
2012 Romania National Olympiad
Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by



$$V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .$$Determine the following two sets:









a) $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$

b) $\{V(f)\, |\, f\in\mathcal{C}\}\ .$
2012 Romania National Olympiad
 Let $m$ and $n$ be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial $\prod_{k=1}^m\, (f+k)$ , when $f$ covers the set of $n^{\text{th}}$ - degree polynomials with complex coefficients.
2011 Romania National Olympiad
Let be a natural number $ n $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that

$$ a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} ,\quad\forall m\in\{ 1,2,\ldots ,n \} . $$Prove that $ a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} . $
2011 Romania National Olympiad
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. $
2011 Romania National Olympiad
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.
2011 Romania National Olympiad
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.
2011 Romania National Olympiad
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.
2011 Romania National Olympiad
Find all numbers $ n $ for which there exist three (not necessarily distinct) roots of unity of order $ n $ whose sum is $ 

1. $
2011 Romania National Olympiad
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] . $
2011 Romania National Olympiad
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. $
2011 Romania National Olympiad
A row of a matrix belonging to $\mathcal{M}_n(\mathbb{C})$ is said to be permutable if no matter how we would permute the entries of that row, the value of the determinant doesn't change. Prove that if a matrix has two permutable rows, then its determinant is equal to $0$ .
2011 Romania National Olympiad
Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .