Navvye/Olympiad-Problems-CSV huggingface dataset:
2008 Portugal MO
Let $d$ be a natural number. Given two natural numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if the $d$ numbers obtained substituting each one of the digits of $M$ by the digit of $N$ which is on the same position are all multiples of $7$. Find all the values of $d$ for which the following condition is valid:

For any two numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if $N$ is a friend of $M$.
2008 Portugal MO
Nelson challenges Telma for the following game:

First Telma takes $2^9$ numbers from the set $\left\{0,1,2,3,\cdots,1024\right\}$, then Nelson takes $2^8$ of the remaining numbers. Then Telma takes $2^7$ numbers and successively, until only two numbers remain. Nelson will have to give Telma the difference between these two numbers in euros. What is the largest amount Telma can win, whatever Nelson's strategy is?
2008 Portugal MO
Let $ABC$ be a right-angled triangle in $A$ such that $AB<AC$. Let $M$ be the midpoint of $BC$ and let $D$ be the intersection of $AC$ with the perpendicular line to $BC$ which passes through $M$. Let $E$ be the intersection point of the parallel line to $AC$ which passes through $M$ with the perpendicular line to $BD$ which passes through $B$. Prove that triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC=60^{\circ}$.
2008 Portugal MO
Let $n$ be a natural number larger than $2$. Vanessa has $n$ piles of jade stones, and all the piles have a different number of stones. Vanessa can distribute the stones from any pile by the other piles and stay with $n-1$ piles with the same number of stones. She also can distribute the stones from any two piles by the other piles and stay with $n-2$ piles with the same number of stones. Find the smallest possible number of jade's stones that the pile with the largest number of stones can have.
2021 Romania National Olympiad
Solve the system in reals: $\frac{4-a}{b}=\frac{5-b}{a}=\frac{10}{a^2+b^2}$.
2021 Romania National Olympiad
Solve the system in reals:

$(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=2022$ and $x+y=\frac{2021}{\sqrt{2022}}$
2021 Romania National Olympiad
Students in a class of $n$ students had to solve $2^{n-1}$ problems on an exam. It turned out that for each pair of distinct problems:

• there is at least one student who has solved both

• there is at least one student who has solved one of them, but not the other.

Show that there is a problem solved by all the students in the class.
2021 Romania National Olympiad
If $a,b,c>0,a+b+c=1$,then:



$\frac{1}{abc}+\frac{4}{a^{2}+b^{2}+c^{2}}\geq\frac{13}{ab+bc+ca}$
2021 Romania National Olympiad
Find the complex numbers $x,y,z$,with $\mid x\mid=\mid y\mid=\mid z\mid$,knowing that



$x+y+z$ and $x^{3}+y^{3}+z^{3}$ are be real numbers.
2021 Romania National Olympiad
Let $f:[a,b] \rightarrow \mathbb{R}$ a function with Intermediate Value property such that $f(a) * f(b) < 0$. Show that there exist $\alpha$, $\beta$ such that $a < \alpha < \beta  < b$ and $f(\alpha) + f(\beta) = f(\alpha) * f(\beta)$.
2021 Romania National Olympiad
Let $n \ge 2$ and  $ a_1, a_2, \ldots , a_n $, nonzero real numbers not  necessarily distinct.  We define matrix $A = (a_{ij})_{1 \le i,j \le n} \in M_n( \mathbb{R} )$ , $a_{i,j} = max \{ a_i, a_j \}$, $\forall i,j \in \{ 1,2 , \ldots , n \} $. Show that $\mathbf{rank}(A) $= $\mathbf{card} $ $\{ a_k | k = 1,2, \ldots n \} $
2021 Romania National Olympiad
Let $f :\mathbb R \to\mathbb R$ a function $ n \geq 2$ times differentiable so that:

$ \lim_{x \to \infty} f(x) = l \in \mathbb R$ and $ \lim_{x \to \infty} f^{(n)}(x) = 0$.

Prove that: $ \lim_{x \to \infty} f^{(k)}(x) = 0 $ for all $ k \in \{1, 2, \dots, n - 1\} $, where $f^{(k)}$ is the $ k $ - th derivative of $f$.
2021 Romania National Olympiad
Let $n \ge 2$ and matrices $A,B \in M_n(\mathbb{R})$. There exist $x \in \mathbb{R} \backslash \{0,\frac{1}{2}, 1 \}$, such that $ xAB + (1-x)BA = I_n$. Show that $(AB-BA)^n = O_n$.
2021 Romania National Olympiad
Find all continuous functions $f:\left[0,1\right]\rightarrow[0,\infty)$ such that:



$\int_{0}^{1}f\left(x\right)dx\cdotp\int_{0}^{1}f^{2}\left(x\right)dx\cdotp...\cdotp\int_{0}^{1}f^{2020}\left(x\right)dx=\left(\int_{0}^{1}f^{2021}\left(x\right)dx\right)^{1010}$
2021 Romania National Olympiad
Given is an positive integer $a>2$

a) Prove that there exists positive integer $n$ different from $1$, which is not a prime, such that $a^n=1(mod n)$

b) Prove that if $p$ is the smallest positive integer, different from $1$, such that $a^p=1(mod p)$, then $p$ is a prime.

c) There does not exist positive integer $n$, different from $1$, such that $2^n=1(mod n)$
2021 Romania National Olympiad
Let be $f:\left[0,1\right]\rightarrow\left[0,1\right]$ a continuous and bijective function,such that :



$f\left(0\right)=0$.Then the following inequality holds:



$\left(\alpha+2\right)\cdotp\int_{0}^{1}x^{\alpha}\left(f\left(x\right)+f^{-1}\left(x\right)\right)\leq2,\forall\alpha\geq0 $
2019 Romania National Olympiad
Let be a point $ P $ in the interior of a triangle $ ABC $ such that $ BP=AC, M $ be the middlepoint of the segment $ AP,  

 R $ be the middlepoint of $ BC $ and $ E $ be the intersection of $ BP $ with $ AC. $ Prove that the bisector of $ \angle BEA $ is perpendicular on $ MR $
2019 Romania National Olympiad
Find all natural numbers which are the cardinal of a set of nonzero Euclidean vectors whose sum is $ 0, $ the sum of any two of them is nonzero, and their magnitudes are equal.
2019 Romania National Olympiad
Prove that the number of solutions in $ \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right) $ of the parametric equation

$$ \sqrt{x^2+y+n}+\sqrt{y^2+x+n} = z, $$is greater than zero and finite, for nay natural number $ n. $
2019 Romania National Olympiad
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x+y)\leq f(x^2+y)$$for all $x,y$.
2019 Romania National Olympiad
If $a,b,c>0$ then

$$\frac{1}{abc}+1\ge3\left(\frac{1}{a^2+b^2+c^2}+\frac{1}{a+b+c}\right)$$
2019 Romania National Olympiad
Find the number of trapeziums that it can be formed with the vertices of a regular polygon.
2019 Romania National Olympiad
Find all natural numbers $ n\ge 4 $ that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers $ a,b,c $ that verify the equation

$$ (a-b)^n+(b-c)^n+(c-a)^n=0, $$represent the vertices of an equilateral triangle in the complex plane.
2019 Romania National Olympiad
Let be two nonempty finite sets $ A,B $ consisting of nonegative integers. Prove that

$$ \left\{ f:\mathcal{P} (A)\longrightarrow B| f\left( X\cap Y \right) =\min\left( \left| f(X) \right| ,\left| f(Y) \right| \right) \right\} =\left\{ g:\mathcal{P} (A)\longrightarrow B| f\left( X\cup Y \right) =\max\left( \left| g(X) \right| ,\left| g(Y) \right| \right) \right\} $$and find the cardinal of these sets.



$ \mathcal{P} $ denotes the power of sets.
2019 Romania National Olympiad
Let $n \geq 2$ and $A, B \in \mathcal{M}_n(\mathbb{C})$ such that there exists an idempotent matrix $C \in \mathcal{M}_n(\mathbb{C})$ for which $C^*=AB-BA.$ Prove that $(AB-BA)^2=0.$



Note: $X^*$ is the adjugate matrix of $X$ (not the conjugate transpose)
2019 Romania National Olympiad
Let $f:[0, \infty) \to \mathbb{R}$ a continuous function, constant on $\mathbb{Z}_{\geq 0}.$ For any $0 \leq a < b < c < d$ which satisfy $f(a)=f(c)$ and $f(b)=f(d)$ we also have $f \left( \frac{a+b}{2} \right) = f \left( \frac{c+d}{2} \right).$

Prove that $f$ is constant.
2019 Romania National Olympiad
$\textbf{a)}$ Prove that there exists a differentiable function $f:(0, \infty) \to (0, \infty)$ such that $f(f'(x)) = x, \: \forall x>0.$

$\textbf{b)}$ Prove that there is no differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f'(x)) = x, \: \forall x \in \mathbb{R}.$
2019 Romania National Olympiad
Let $p$ be a prime number. For any $\sigma \in S_p$ (the permutation group of $\{1,2,...,p \}),$ define the matrix $A_{\sigma}=(a_{ij}) \in \mathcal{M}_p(\mathbb{Z})$ as $a_{ij} = \sigma^{i-1}(j),$ where $\sigma^0$ is the identity permutation and $\sigma^k = \underbrace{\sigma \circ \sigma \circ ... \circ \sigma}_k.$

Prove that $D = \{ |\det A_{\sigma}| : \sigma \in S_p \}$ has at most $1+ (p-2)!$ elements.
2019 Romania National Olympiad
Let $a>0$ and $\mathcal{F} = \{f:[0,1] \to \mathbb{R} : f \text{ is concave and } f(0)=1 \}.$ Determine $$\min_{f \in \mathcal{F}} \bigg\{ \left( \int_0^1 f(x)dx\right)^2 - (a+1) \int_0^1 x^{2a}f(x)dx \bigg\}.$$
2019 Romania National Olympiad
Let $n \geq 4$ be an even natural number and $G$ be a subgroup of $GL_2(\mathbb{C})$ with $|G| = n.$ Prove that there exists $H \leq G$ such that $\{ I_2 \} \neq H$ and $H \neq G$ such that $XYX^{-1} \in H, \: \forall X \in G$ and $\forall Y \in H$
2019 Romania National Olympiad
Let $f:[0, \infty) \to (0, \infty)$ be an increasing function and $g:[0, \infty) \to \mathbb{R}$ be a two times differentiable function such that $g''$ is continuous and $g''(x)+f(x)g(x) = 0, \: \forall x \geq 0.$

$\textbf{a)}$ Provide an example of such functions, with $g \neq 0.$

$\textbf{b)}$ Prove that $g$ is bounded.
2019 Romania National Olympiad
Let $n \geq 3$ and $a_1,a_2,...,a_n$ be complex numbers different from $0$ with $|a_i| < 1$ for all $i \in \{1,2,...,n-1 \}.$ If the coefficients of $f = \prod_{i=1}^n (X-a_i)$ are integers, prove that

$\textbf{a)}$ The numbers $a_1,a_2,...,a_n$ are distinct.

$\textbf{b)}$ If $a_j^2 = a_ia_k,$ then $i=j=k.$
2018 Romania National Olympiad
Prove that if in a triangle the orthocenter, the centroid and the incenter are collinear, then the triangle is isosceles.
2018 Romania National Olympiad
Let $a,b,c \geq 0$ and $a+b+c=3.$ Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \geq \frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+a}$$
2018 Romania National Olympiad
Let $f,g : \mathbb{R} \to \mathbb{R}$ be two quadratics such that, for any real number $r,$ if $f(r)$ is an integer, then $g(r)$ is also an integer.

Prove that there are two integers $m$ and $n$ such that $$g(x)=mf(x)+n, \: \forall x \in \mathbb{R}$$
2018 Romania National Olympiad
Let $n \in \mathbb{N}^*$ and consider a circle of length $6n$ along with $3n$ points on the circle which divide it into $3n$ arcs: $n$ of them have length $1,$ some other $n$ have length $2$ and the remaining $n$ have length $3.$

Prove that among these points there must be two such that the line that connects them passes through the center of the circle.
2018 Romania National Olympiad
Let $n \in \mathbb{N}_{\geq 2}$ and $a_1,a_2, \dots , a_n \in (1,\infty).$ Prove that $f:[0,\infty) \to \mathbb{R}$ with $$f(x)=(a_1a_2...a_n)^x-a_1^x-a_2^x-...-a_n^x$$is a strictly increasing function.
2018 Romania National Olympiad
Let $ABC$ be a triangle, $O$ its circumcenter and $R=1$ its circumradius. Let $G_1,G_2,G_3$ be the centroids of the triangles $OBC, OAC$ and $OAB.$ Prove that the triangle $ABC$ is equilateral if and only if $$AG_1+BG_2+CG_3=4$$
2018 Romania National Olympiad
Let $n \in \mathbb{N}_{\geq 2}.$ Prove that for any complex numbers $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n,$ the following statements are equivalent:

a) $\sum_{k=1}^n|z-a_k|^2 \leq \sum_{k=1}^n|z-b_k|^2, \: \forall z \in \mathbb{C}.$

b) $\sum_{k=1}^na_k=\sum_{k=1}^nb_k$ and $\sum_{k=1}^n|a_k|^2 \leq \sum_{k=1}^n|b_k|^2.$
2018 Romania National Olympiad
Let $n \in \mathbb{N}_{\geq 2}.$ For any real numbers $a_1,a_2,...,a_n$ denote $S_0=1$ and for $1 \leq k \leq n$ denote

$$S_k=\sum_{1 \leq i_1 < i_2 < ... <i_k \leq n}a_{i_1}a_{i_2}...a_{i_k}$$Find the number of $n-$tuples $(a_1,a_2,...a_n)$ such that $$(S_n-S_{n-2}+S_{n-4}-...)^2+(S_{n-1}-S_{n-3}+S_{n-5}-...)^2=2^nS_n.$$
2018 Romania National Olympiad
Let $n \geq 2$ be a positive integer and, for all vectors with integer entries $$X=\begin{pmatrix}

x_1 \\
x_2 \\
\vdots \\
x_n

\end{pmatrix}$$let $\delta(X) \geq 0$ be the greatest common divisor of $x_1,x_2, \dots, x_n.$ Also, consider $A \in \mathcal{M}_n(\mathbb{Z}).$

Prove that the following statements are equivalent:

$\textbf{i) }$ $|\det A | = 1$

$\textbf{ii) }$ $\delta(AX)=\delta(X),$ for all vectors $X \in \mathcal{M}_{n,1}(\mathbb{Z}).$



Romeo Raicu
2018 Romania National Olympiad
Let $x>0.$ Prove that $$2^{-x}+2^{-1/x} \leq 1.$$
2018 Romania National Olympiad
Let $f: \mathbb{R} \to \mathbb{R}$ be a function with the intermediate value property. If $f$ is injective on $\mathbb{R} \setminus \mathbb{Q},$ prove that $f$ is continuous on $\mathbb{R}.$



Julieta R. Vergulescu
2018 Romania National Olympiad
Let $n$ be an integer with $n \geq 2$ and let $A \in \mathcal{M}_n(\mathbb{C})$ such that $\operatorname{rank} A \neq \operatorname{rank} A^2.$ Prove that there exists a nonzero matrix $B \in \mathcal{M}_n(\mathbb{C})$ such that $$AB=BA=B^2=0$$

Cornel Delasava
2018 Romania National Olympiad
Let $A$ be a finite ring and $a,b \in A,$ such that $(ab-1)b=0.$ Prove that $b(ab-1)=0.$
2018 Romania National Olympiad
Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}$$For $f \in \mathcal{F},$ let $$I(f)=\int_0^ef(x) dx$$Determine $\min_{f \in \mathcal{F}}I(f).$



Liviu Vlaicu
2018 Romania National Olympiad
Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$

$\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$

$\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$

Nicolae Bourbacut
2018 Romania National Olympiad
For any $k \in \mathbb{Z},$ define $$F_k=X^4+2(1-k)X^2+(1+k)^2.$$Find all values $k \in \mathbb{Z}$ such that $F_k$ is irreducible over $\mathbb{Z}$ and reducible over $\mathbb{Z}_p,$ for any prime $p.$



Marius Vladoiu
2017 Romania National Olympiad
Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.
2017 Romania National Olympiad
Let be a square $ ABCD, $ a point $ E $ on $ AB, $ a point $ N $ on $ CD, $ points $ F,M $ on $ BC, $ name $ 

P $ the intersection of $ AN $ with $ DE, $ and name $ Q $ the intersection of $ AM $ with $ EF. $ If the triangles $ AMN $ and $ DEF $ are equilateral, prove that $ PQ=FM. $
2017 Romania National Olympiad
Let be two natural numbers $ n $ and $ a. $

a) Prove that there exists an $ n\text{-tuplet} $ of natural numbers $ \left( a_1,a_2,\ldots ,a_n\right) $ that satisfy the following equality.

$$ 1+\frac{1}{a} =\prod_{i=1}^n \left( 1+\frac{1}{a_i} \right) $$b) Show that there exist only finitely such $ n\text{-tuplets} . $
2017 Romania National Olympiad
Let be two natural numbers $ b>a>0 $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property.

$$ f\left( x^2+ay\right)\ge f\left( x^2+by\right) ,\quad\forall x,y\in\mathbb{R} $$

a) Show that $ f(s)\le f(0)\le f(t) , $ for any real numbers $ s<0<t. $

b) Prove that $ f $ is constant on the interval $ (0,\infty ) . $

c) Give an example of a non-monotone such function.
2017 Romania National Olympiad
Solve in the set of real numbers the equation $ a^{[ x ]} +\log_a\{ x \} =x , $ where $ a $ is a real number from the interval $ (0,1). $



$ [] $ and $ \{\} $ denote the floor, respectively, the fractional part.
2017 Romania National Olympiad
A function $ f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q} $ has the following property:

$$ f(xy)=f(x)+f(y),\quad x,y\in\mathbb{Q}_{>0} $$

a) Demonstrate that there are no injective functions with this property.

b) Do exist surjective functions having this property?
2017 Romania National Olympiad
$ \sin\frac{\pi }{4n}\ge \frac{\sqrt 2 }{2n} ,\quad \forall n\in\mathbb{N} $
2017 Romania National Olympiad
Find the number of functions $ A\stackrel{f}{\longrightarrow } A $ for which there exist two functions $ A\stackrel{g}{\longrightarrow } B\stackrel{h}{\longrightarrow } A $ having the properties that $ g\circ h =\text{id.} $ and $ h\circ g=f, $ where $ B $ and $ A $ are two finite sets.
2017 Romania National Olympiad
Let be a surjective function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that if the sequence $ \left( f\left( x_n \right) \right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. Prove that it is continuous.
2017 Romania National Olympiad
Let be two natural numbers $ n\ge 2, k, $ and $ k\quad n\times n $ symmetric real matrices $ A_1,A_2,\ldots ,A_k. $ Then, the following relations are equivalent:



$ 1)\quad \left| \sum_{i=1}^k A_i^2 \right| =0 $



$ 2)\quad \left| \sum_{i=1}^k A_iB_i \right| =0,\quad\forall B_1,B_2,\ldots ,B_k\in \mathcal{M}_n\left( \mathbb{R} \right) $



$ || $ denotes the determinant.
2017 Romania National Olympiad
Let be a natural number $ n\ge 2 $ and two $ n\times n $ complex matrices $ A,B $ that satisfy $ (AB)^3=O_n. $

Does this imply that $ (BA)^3=O_n ? $
2017 Romania National Olympiad
Let be a function $ f $ of class $ \mathcal{C}^1[a,b] $ whose derivative is positive. Prove that there exists a real number $ c\in (a,b) $ such that

$$ f(f(b))-f(f(a))=(f'(c))^2(b-a) . $$
2017 Romania National Olympiad
a) Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation.

$$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$

b) Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.
2017 Romania National Olympiad
Let be a natural number $ n $ and $ 2n $ real numbers $ b_1,b_2,\ldots ,b_n,a_1<a_2<\cdots <a_n. $ Show that



a) if $ b_1,b_2,\ldots ,b_n>0, $ then there exists a polynomial $ f\in\mathbb{R}[X] $ irreducible in $ \mathbb{R}[X] $ such that $$ f\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$b) there exists a polynom $ g\in\mathbb{R} [X] $ of degree at least $ 1 $ which has only real roots and such that

$$ g\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$
2017 Romania National Olympiad
Let $G$ be a finite group with the following property:



If $f$ is an automorphism of $G$, then there exists $m\in\mathbb{N^\star}$, so that $f(x)=x^{m} $ for all $x\in G$.



Prove that G is commutative.



Marian Andronache
2017 Romania National Olympiad
A function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ has the property that $ \lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1. $



a) Give an example of what $ f $ could be if it's continuous and $ f/\text{id.} $ doesn't have a limit at $ \infty . $

b) Prove that if $ f $ is nondecreasing then $ f/\text{id.} $ has a limit at $ \infty , $ and determine it.
2016 Romania National Olympiad
The orthocenter $ H $ of a triangle $ ABC $ is distinct from its vertices and its circumcenter $ O. $ $ M,N,P $ are the circumcenters of the triangles $ HBC,HCA, $ respectively, $ HAB. $ Prove that $ AM,BN,CP $ and $ OH $ are concurrent.
2016 Romania National Olympiad
Let be a natural number $ n\ge 2 $ and $ n $ positive real numbers $ a_1,a_n,\ldots ,a_n $ that satisfy the inequalities

$$ \sum_{j=1}^i a_j\le a_{i+1} ,\quad \forall i\in\{ 1,2,\ldots ,n-1 \} . $$Prove that

$$ \sum_{k=1}^{n-1} \frac{a_k}{a_{k+1}}\le n/2 . $$
2016 Romania National Olympiad
We say that a rational number is spheric if it is the sum of three squares of rational numbers (not necessarily distinct). Prove that:



a) $ 7 $ is not spheric.

b) a rational spheric number raised to the power of any natural number greater than $ 1 $ is spheric.
2016 Romania National Olympiad
Determine all functions $f: \mathbb R \to \mathbb R$ which satisfy the inequality

$$f(a^2) - f(b^2) \leq \left( f(a) + b\right)\left( a - f(b)\right),$$for all $a,b \in \mathbb R$.
2016 Romania National Olympiad
Let be a natural number $ n\ge 2 $ and $ n $ positive real numbers $ a_1,a_2,\ldots ,a_n $ whose product is $ 1. $

Prove that the function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R} ,\quad f(x)=\prod_{i=1}^n \left( 1+a_i^x \right) , $ is nondecreasing.
2016 Romania National Olympiad
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying the conditions:

$$ \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\   f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , $$for all real numbers $ x,y,t $ with $ t\in [0,1] . $



Prove that:

a) $ f(b)+f(c)\le f(a)+f(d) , $ for any real numbers $ a,b,c,d $ such that $ a\le b\le c\le d $ and $ d-c=b-a. $

b) for any natural number $ n\ge 3 $ and any $ n $ real numbers $ x_1,x_2,\ldots ,x_n, $ the following inequality holds.

$$ f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right) $$
2016 Romania National Olympiad
a) Let be two nonzero complex numbers $ a,b. $ Show that the area of the triangle formed by the representations of the affixes $ 0,a,b $ in the complex plane is $ \frac{1}{4}\left| \overline{a} b-a\overline{b} \right| . $



b) Let be an equilateral triangle $ ABC, $ its circumcircle $ \mathcal{C} , $ its circumcenter $ O, $ and two distinct points $ P_1,P_2 $ in the interior of $ \mathcal{C} . $ Prove that we can form two triangles with sides $ P_1A,P_1B,P_1C, $ respectively, $ P_2A,P_2B,P_2C, $ whose areas are equal if and only if $ OP_1=OP_2. $
2016 Romania National Olympiad
In order to study a certain ancient language, some researchers formatted its discovered words into expressions formed by concatenating letters from an alphabet containing only two letters. Along the study, they noticed that any two distinct words whose formatted expressions have an equal number of letters, greater than $ 2, $ differ by at least three letters.

Prove that if their observation holds indeed, then the number of formatted expressions that have $ n\ge 3 $ letters is at most $ \left[ \frac{2^n}{n+1} \right] . $
2016 Romania National Olympiad
Let be a $ 2\times 2 $ real matrix $ A $ that has the property that $ \left| A^d-I_2 \right| =\left| A^d+I_2 \right| , $ for all $ d\in\{ 2014,2016 \} . $

Prove that $ \left| A^n-I_2 \right| =\left| A^n+I_2 \right| , $ for any natural number $ n. $
2016 Romania National Olympiad
Consider a natural number, $ n\ge 2, $ and three $ n\times n $ complex matrices $ A,B,C $ such that $ A $ is invertible, $ B $ is formed by replacing the first line of $ A $ with zeroes, and $ C $ is formed by putting the last $ n-1 $ lines of $ A $ above a line of zeroes. Prove that:



a) $ \text{rank} \left( A^{-1} B \right) = \text{rank} \left( \left( A^{-1} B\right)^2 \right) =\cdots =\text{rank} \left( \left( A^{-1} B\right)^n \right) $

b) $ \text{rank} \left( A^{-1} C \right) > \text{rank} \left( \left( A^{-1} C\right)^2 \right) >\cdots >\text{rank} \left( \left( A^{-1} C\right)^n \right) $
2016 Romania National Olympiad
Let be a real number $ a, $ and a function $ f:\mathbb{R}_{>0 }\longrightarrow\mathbb{R}_{>0 } . $ Show that the following relations are equivalent.



$ \text{(i)}\quad\varepsilon\in\mathbb{R}_{>0 } \implies\left( \lim_{x\to\infty } \frac{f(x)}{x^{a+\varepsilon }} =0\wedge \lim_{x\to\infty } \frac{f(x)}{x^{a-\varepsilon }} =\infty \right) $

$ \text{(ii)}\quad\lim_{x\to\infty } \frac{\ln f(x)}{\ln x } =a $
2016 Romania National Olympiad
Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $
2016 Romania National Olympiad
Prove that there exists an unique sequence $ \left( c_n \right)_{n\ge 1} $ of real numbers from the interval $ (0,1) $ such that$$ \int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } , $$for all natural numbers $ m, $ and calculate $ \lim_{k\to\infty } kc_k^k. $





Radu Pop