Navvye/Olympiad-Problems-CSV huggingface dataset:
2004 Romania National Olympiad
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq |x-y|$, for all $x,y \in \mathbb{R}$.



Prove that if for any real $x$, the sequence $x,f(x),f(f(x)),\ldots$ is an arithmetic progression, then there is $a \in \mathbb{R}$ such that  $f(x)=x+a$, for all $x \in \mathbb R$.
2004 Romania National Olympiad
Let $ABCD$ be a tetrahedron in which the opposite sides are equal and form  equal angles.



Prove that it is regular.
2004 Romania National Olympiad
Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: $$ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . $$



Radu Gologan
2004 Romania National Olympiad
Let $\displaystyle \left( P_n \right)_{n \geq 1}$ be an infinite family of planes and $\displaystyle \left( X_n \right)_{n \geq 1}$ be a family of non-void, finite sets of points such that $\displaystyle X_n \subset P_n$ and the projection of the set $\displaystyle X_{n+1}$ on the plane $\displaystyle P_n$ is included in the set $X_n$, for all $n$.



Prove that there is a sequence of points $\displaystyle \left( p_n \right)_{n \geq 1}$ such that $\displaystyle p_n \in P_n$  and $p_n$ is the projection of $p_{n+1}$ on the plane $P_n$, for all $n$.



Does the conclusion of the problem remain true if the sets $X_n$ are infinite?



Claudiu Raicu
2004 Romania National Olympiad
Let $n \geq 3$ be an integer and $F$ be the focus of the parabola $y^2=2px$. A regular polygon $A_1 A_2 \ldots A_n$ has the center in $F$ and none of its vertices lie on $Ox$. $\left( FA_1 \right., \left( FA_2 \right., \ldots, \left( FA_n \right.$ intersect the parabola at $B_1,B_2,\ldots,B_n$.



Prove that $$ FB_1 + FB_2 + \ldots + FB_n > np .  $$



Calin Popescu
2004 Romania National Olympiad
Let $n \in \mathbb N$, $n \geq 2$.



(a) Give an example of two matrices $A,B \in \mathcal M_n \left( \mathbb C \right)$ such that $$ \textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor .  $$



(b) Prove that for all matrices $X,Y \in \mathcal M_n \left( \mathbb C \right)$ we have $$ \textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor .  $$



Ion Savu
2004 Romania National Olympiad
Let $f : (a,b) \to \mathbb R$ be a function with the property that for all $x \in (a,b)$ there is a non-degenerated interval $[ a_x,b_x ]$ with $a < a_x \leq x \leq b_x < b$ such that $f$ is constant on $\left[ a_x,b_x \right]$.



(a) Prove that $\textrm{Im} \, f$ is finite or numerable.



(b) Find all continuous functions which have the property mentioned in the hypothesis.
2004 Romania National Olympiad
(a) Build a function $f : \mathbb R \to \mathbb R_+$ with the property $\left( \mathcal P \right)$, i.e. all $x \in \mathbb Q$ are local, strict minimum points.



(b) Build a function $f : \mathbb Q \to \mathbb R_+$ such that every point is a local, strict minimum point and such that $f$ is unbounded on $I \cap \mathbb Q$, where $I$ is a non-degenerate interval.



(c) Let $f: \mathbb R \to \mathbb R_+$ be a function unbounded on every $I \cap \mathbb Q$, where $I$ is a non-degenerate interval. Prove that $f$ doesn't have the property $\left( \mathcal P \right)$.
2004 Romania National Olympiad
Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have $$ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 .  $$



Mihai Piticari
2004 Romania National Olympiad
Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$.



(a) Prove that $f_n$ is well defined.



(b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective.



Bogdan Enescu
2004 Romania National Olympiad
Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that $$ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 .  $$ Prove that $$ \int_0^1 f^2 (x) \, dx \geq 4 .  $$

Ion Rasa
2004 Romania National Olympiad
Let $\mathcal K$ be a field of characteristic $p$, $p \equiv 1 \left( \bmod 4 \right)$.



(a) Prove that $-1$ is the square of an element from $\mathcal K.$



(b) Prove that any element $\neq 0$ from $\mathcal K$ can be written as the sum of three squares, each $\neq 0$, of elements from $\mathcal K$.



(c) Can $0$ be written in the same way?



Marian Andronache
2003 Romania National Olympiad
Find the maximum number of elements which can be chosen from the set $ \{1,2,3,\ldots,2003\}$ such that the sum of any two chosen elements is not divisible by 3.
2003 Romania National Olympiad
Compute the maximum area of a triangle having a median of length 1 and a median of length 2.
2003 Romania National Olympiad
For every positive integer $ n$ consider

$$ A_n=\sqrt{49n^2+0,35n}.

$$

(a) Find the first three digits after decimal point of $ A_1$.

(b) Prove that the first three digits after decimal point of $ A_n$ and $ A_1$ are the same, for every $ n$.
2003 Romania National Olympiad
In triangle $ ABC$, $ P$ is the midpoint of side $ BC$. Let $ M\in(AB)$, $ N\in(AC)$ be such that $ MN\parallel BC$ and $ \{Q\}$ be the common point of $ MP$ and $ BN$. The perpendicular from $ Q$ on $ AC$ intersects $ AC$ in $ R$ and the parallel from $ B$ to $ AC$ in $ T$. Prove that:

(a) $ TP\parallel MR$;

(b) $ \angle MRQ=\angle PRQ$.



Mircea Fianu
2003 Romania National Olympiad
Let $ m,n$ be positive integers. Prove that the number $ 5^n+5^m$ can be represented as sum of two perfect squares if and only if $ n-m$ is even.



Vasile Zidaru
2003 Romania National Olympiad
In a meeting there are 6 participants. It is known that among them there are seven pairs of friends and in any group of three persons there are at least two friends. Prove that:

(a) there exists a person who has at least three friends;

(b) there exists three persons who are friends with each other.



Valentin Vornicu
2003 Romania National Olympiad
The real numbers $ a,b$ fulfil the conditions

(i) $ 0<a<a+\frac12\le b$;

(ii) $ a^{40}+b^{40}=1$.

Prove that $ b$ has the first 12 digits after the decimal point equal to 9.



Mircea Fianu
2003 Romania National Olympiad
In tetrahedron $ ABCD$, $ G_1,G_2$ and $ G_3$ are barycenters of the faces $ ACD,ABD$ and $ BCD$ respectively.

(a) Prove that the straight lines $ BG_1,CG_2$ and $ AG_3$ are concurrent.

(b) Knowing that $ AG_3=8,BG_1=12$ and $ CG_2=20$ compute the maximum possible value of the volume of $ ABCD$.
2003 Romania National Olympiad
Find positive integers $ a,b$ if for every $ x,y\in[a,b]$, $ \frac1x+\frac1y\in[a,b]$.
2003 Romania National Olympiad
An integer $ n$, $ n\ge2$ is called friendly if there exists a family $ A_1,A_2,\ldots,A_n$ of subsets of the set $ \{1,2,\ldots,n\}$ such that:

(1) $ i\not\in A_i$ for every $ i=\overline{1,n}$;

(2) $ i\in A_j$ if and only if $ j\not\in A_i$, for every distinct $ i,j\in\{1,2,\ldots,n\}$;

(3) $ A_i\cap A_j$ is non-empty, for every $ i,j\in\{1,2,\ldots,n\}$.

Prove that:

(a) 7 is a friendly number;

(b) $ n$ is friendly if and only if $ n\ge7$.



Valentin Vornicu
2003 Romania National Olympiad
Prove that the midpoints of the altitudes of a triangle are collinear if and only if the triangle is right.



Dorin Popovici
2003 Romania National Olympiad
Let $ P$ be a plane. Prove that there exists no function $ f: P\rightarrow P$ such that for every convex quadrilateral $ ABCD$, the points $ f(A),f(B),f(C),f(D)$ are the vertices of a concave quadrilateral.



Dinu Şerbănescu
2003 Romania National Olympiad
Let be a tetahedron $ OABC $ with $ OA\perp OB\perp OC\perp OA. $ Show that

$$ OH\le r\left( 1+\sqrt 3 \right) , $$where $ H $ is the orthocenter of $ ABC $ and $ r $ is radius of the inscribed spere of $ OABC. $





Valentin Vornicu
2003 Romania National Olympiad
Let be five nonzero complex numbers having the same absolute value and such that zero is equal to their sum, which is equal to the sum of their squares. Prove that the affixes of these numbers in the complex plane form a regular pentagon.





Daniel Jinga
2003 Romania National Olympiad
Let be a circumcircle of radius $ 1 $ of a triangle whose centered representation in the complex plane is given by the affixes of $ a,b,c, $ and for which the equation $ a+b\cos x +c\sin x=0 $ has a real root in $ \left( 0,\frac{\pi }{2} \right) . $ prove that the area of the triangle is a real number from the interval $ \left( 1,\frac{1+\sqrt 2}{2} \right] . $





Gheorghe Iurea
2003 Romania National Olympiad
a) Prove that the sum of all the elements of a finite union of sets of elements of finite cyclic subgroups of the group of complex numbers, is an integer number.

b) Show that there are finite union of sets of elements of finite cyclic subgroups of the group of complex numbers such that the sum of all its elements is equal to any given integer.





Paltin Ionescu
2003 Romania National Olympiad
Find the locus of the points $ M $ that are situated on the plane where a rhombus $ ABCD $ lies, and satisfy:

$$ MA\cdot MC+MB\cdot MD=AB^2 $$



Ovidiu Pop
2003 Romania National Olympiad
Let be eight real numbers $ 1\le a_1\le a_2\le a_3\le a_4,x_1<x_2<x_3<x_4.  $ Prove that

$$ \begin{vmatrix}a_1^{x_1} & a_1^{x_2} & a_1^{x_3} & a_1^{x_4} \\
a_2^{x_1} & a_2^{x_2} & a_2^{x_3} & a_2^{x_4} \\
a_3^{x_1} & a_3^{x_2} & a_3^{x_3} & a_3^{x_4} \\
a_4^{x_1} & a_4^{x_2} & a_4^{x_3} & a_4^{x_4} \\
\end{vmatrix} >0. $$



Marian Andronache, Ion Savu
2003 Romania National Olympiad
Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1}  $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) .  $





Radu Gologan
2003 Romania National Olympiad
Let be a $ 3\times 3 $ real matrix $ A. $ Prove the following statements.

a) $ f(A)\neq O_3, $ for any polynomials $ f\in\mathbb{R} [X] $ whose roots are not real.

b) $ \exists n\in\mathbb{N}\quad \left( A+\text{adj} (A) \right)^{2n} =\left( A \right)^{2n} +\left( \text{adj} (A) \right)^{2n}\iff \text{det} (A)=0 $





Laurențiu Panaitopol
2003 Romania National Olympiad
a) Determine the center of the ring of square matrices of a certain dimensions with elements in a given field, and prove that it is isomorphic with the given field.

b) Prove that

$$ \left(\mathcal{M}_n\left( \mathbb{R} \right) ,+, \cdot\right)\not\cong \left(\mathcal{M}_n\left( \mathbb{C} \right) ,+,\cdot\right) , $$for any natural number $ n\ge 2. $



Marian Andronache, Ion Sava
2003 Romania National Olympiad
Let be an odd natural number $ n\ge 3. $ Find all continuous functions $ f:[0,1]\longrightarrow\mathbb{R} $ that satisfy the following equalities.

$$ \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\} $$



Titu Andreescu
2003 Romania National Olympiad
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that

$$ xf(x)\ge \int_0^x f(t)dt , $$for all real numbers $ x. $ Prove that



a) the mapping $ x\mapsto \frac{1}{x}\int_0^x  f(t) dt $ is nondecreasing on the restrictions $ \mathbb{R}_{<0 } $ and  $ \mathbb{R}_{>0 } . $



b) if $ \int_x^{x+1} f(t)dt=\int_{x-1}^x f(t)dt , $ for any real number $ x, $ then $ f $ is constant.





Mihai Piticari
2003 Romania National Olympiad
$ i(L) $ denotes the number of multiplicative binary operations over the set of elements of the finite additive group $ L $ such that the set of elements of $ L, $ along with these additive and multiplicative operations, form a ring. Prove that



a) $ i\left( \mathbb{Z}_{12} \right) =4. $

b) $ i(A\times B)\ge i(A)i(B) , $ for any two finite commutative groups $ B $ and $ A. $

c) there exist two sequences $ \left( G_k \right)_{k\ge 1} ,\left( H_k \right)_{k\ge 1} $ of finite commutative groups such that

$$ \lim_{k\to\infty }\frac{\# G_k }{i\left( G_k \right)} =0 $$and

$$ \lim_{k\to\infty }\frac{\# H_k }{i\left( H_k \right)} =\infty. $$



Barbu Berceanu
2002 Romania National Olympiad
Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours have altogether an odd number of winning cards.

Show that any player has at that moment at least one winning card.
2002 Romania National Olympiad
Prove that any real number $0<x<1$ can be written as a difference of two positive and less than $1$ irrational numbers.
2002 Romania National Olympiad
Let $ABCD$ be a trapezium and $AB$ and $CD$ be it's parallel edges. Find, with proof, the set of interior points $P$ of the trapezium which have the property that $P$ belongs to at least two lines each intersecting the segments $AB$ and $CD$ and each dividing the trapezium in two other trapezoids with equal areas.
2002 Romania National Olympiad
$a)$ An equilateral triangle of sides $a$ is given and a triangle $MNP$ is constructed under the following conditions: $P\in (AB),M\in (BC),N\in (AC)$, such that $MP\perp AB,NM\perp BC$ and $PN\perp AC$. Find the length of the segment $MP$.

$b)$ Show that for any acute triangle $ABC$ one can find points $P\in (AB),M\in (BC),N\in (AC)$ such that $MP\perp AB,NM\perp BC$ and $PN\perp AC$.
2002 Romania National Olympiad
For any number $n\in\mathbb{N},n\ge 2$, denote by $P(n)$ the number of pairs $(a,b)$ whose elements are of positive integers such that

$$\frac{n}{a}\in (0,1),\quad \frac{a}{b}\in (1,2)\quad \text{and}\quad \frac{b}{n}\in (2,3). $$

$a)$ Calculate $P(3)$.

$b)$ Find $n$ such that $P(n)=2002$.
2002 Romania National Olympiad
Given real numbers $a,c,d$ show that there exists at most one function $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies:

$$f(ax+c)+d\le x\le f(x+d)+c\quad\text{for any}\ x\in\mathbb{R}$$
2002 Romania National Olympiad
Let $[ABCDEF]$ be a frustum of a regular pyramid. Let $G$ and $G'$ be the centroids of bases $ABC$ and $DEF$ respectively. It is known that $AB=36,DE=12$ and $GG'=35$.

$a)$ Prove that the planes $(ABF),(BCD),(CAE)$ have a common point $P$, and the planes $(DEC),(EFA),(FDB)$ have a common point $P'$, both situated on $GG'$.

$b)$ Find the length of the segment $[PP']$.
2002 Romania National Olympiad
The right prism $[A_1A_2A_3\ldots A_nA_1'A_2'A_3'\ldots A_n'],n\in\mathbb{N},n\ge 3$, has a convex polygon as its base. It is known that $A_1A_2'\perp A_2A_3',A_2A_3'\perp A_3A_4',$$\ldots A_{n-1}A_n'\perp A_nA_1', A_nA_1'\perp A_1A_2'$. Show that:

$a)$ $n=3$;

$b)$ the prism is regular.
2002 Romania National Olympiad
Let $ab+bc+ca=1$. Show that

$$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\sqrt{3}+\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}$$
2002 Romania National Olympiad
Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$. It is known that the symmetric point of $M$with respect to the line $GI$ lies on $AC$. Find the measure of $\measuredangle B$.
2002 Romania National Olympiad
Let $k$ and $n$ be positive integers with $n>2$. Show that the equation:

$$x^n-y^n=2^k$$

has no positive integer solutions.
2002 Romania National Olympiad
Find all functions $f: \mathbb{N}\to\mathbb{N}$ which satisfy the inequality:

$$f(3x+2y)=f(x)f(y)$$

for all non-negative integers $x,y$.
2002 Romania National Olympiad
Let $X,Y,Z,T$ be four points in the plane. The segments $[XY]$ and $[ZT]$ are said to be connected, if there is some point $O$ in the plane such that the triangles $OXY$ and $OZT$ are right-angled at $O$ and isosceles.

Let $ABCDEF$ be a convex hexagon  such that the pairs of segments $[AB],[CE],$ and $[BD],[EF]$ are connected. Show that the points $A,C,D$ and $F$ are the vertices of a parallelogram and $[BC]$ and $[AE]$ are connected.
2002 Romania National Olympiad
Find all real polynomials $f$ and $g$, such that:

$$(x^2+x+1)\cdot f(x^2-x+1)=(x^2-x+1)\cdot g(x^2+x+1), $$

for all $x\in\mathbb{R}$.
2002 Romania National Olympiad
Find all real numbers $a,b,c,d,e$ in the interval $[-2,2]$, that satisfy:

\begin{align*}a+b+c+d+e &= 0\\ a^3+b^3+c^3+d^3+e^3&= 0\\ a^5+b^5+c^5+d^5+e^5&=10 \end{align*}
2002 Romania National Olympiad
Let $I\subseteq \mathbb{R}$ be an interval and $f:I\rightarrow\mathbb{R}$ a function such that:

$$|f(x)-f(y)|\le |x-y|,\quad\text{for all}\ x,y\in I. $$

Show that $f$ is monotonic on $I$ if and only if, for any $x,y\in I$, either $f(x)\le f\left(\frac{x+y}{2}\right)\le f(y)$ or $f(y)\le f\left(\frac{x+y}{2}\right)\le f(x)$.
2002 Romania National Olympiad
In the Cartesian plane consider the hyperbola

$$\Gamma=\{M(x,y)\in\mathbb{R}^2 \vert \frac{x^2}{4}-y^2=1\} $$

and a conic $\Gamma '$, disjoint from $\Gamma$. Let $n(\Gamma ,\Gamma ')$ be the maximal number of pairs of points $(A,A')\in\Gamma\times\Gamma '$ such that $AA'\le BB'$, for any $(B,B')$

For each $p\in\{0,1,2,4\}$, find the equation of $\Gamma'$ for which $n(\Gamma ,\Gamma ')=p$. Justify the answer.
2002 Romania National Olympiad
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has limits at any point and has no local extrema. Show that:

$a)$ $f$ is continuous;

$b)$ $f$ is strictly monotone.
2002 Romania National Olympiad
Let $A\in M_4(C)$ be a non-zero matrix.

$a)$ If $\text{rank}(A)=r<4$, prove the existence of two invertible matrices $U,V\in M_4(C)$, such that:

$$UAV=\begin{pmatrix}I_r&0\\0&0\end{pmatrix}$$

where $I_r$ is the $r$-unit matrix.

$b)$ Show that if $A$ and $A^2$ have the same rank $k$, then the matrix $A^n$ has rank $k$, for any $n\ge 3$.
2002 Romania National Olympiad
Let $f:[0,1]\rightarrow [0,1]$ be a continuous and bijective function.

Describe the set:

$$A=\{f(x)-f(y)\mid x,y\in[0,1]\backslash\mathbb{Q}\}$$

NoteYou are given the result that there is no one-to-one function between the irrational numbers and $\mathbb{Q}$.
2002 Romania National Olympiad
Let $A$ be a ring.

$a)$ Show that the set $Z(A)=\{a\in A|ax=xa,\ \text{for all}\ x\in A\}$ is a subring of the ring $A$.

$b)$ Prove that, if any commutative subring of $A$ is a field, then $A$ is a field.
2002 Romania National Olympiad
Let $f:[0,1]\rightarrow\mathbb{R}$ be an integrable function such that:

$$0<\left\vert 	\int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1.$$

Show that there exists $x_1\not= x_2, x_1,x_2\in [0,1]$, such that:

$$\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}$$
2002 Romania National Olympiad
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and bounded function such that

$$x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t,\quad\text{for any}\ x\in\mathbb{R}.$$

Prove that $f$ is a constant function.
2002 Romania National Olympiad
Let $K$ be a field having $q=p^n$ elements, where $p$ is a prime and $n\ge 2$ is an arbitrary integer number. For any $a\in K$, one defines the polynomial $f_a=X^q-X+a$. Show that:

$a)$ $f=(X^q-X)^q-(X^q-X)$ is divisible by $f_1$;

$b)$ $f_a$ has at least $p^{n-1}$ essentially different irreducible factors $K[X]$.
2001 Romania National Olympiad
Show that there exist no integers $a$ and $b$ such that $a^3+a^2b+ab^2+b^3=2001$.
2001 Romania National Olympiad
Let $a$ and $b$ be real, positive and distinct numbers. We consider the set:

$$M=\{ ax+by\mid x,y\in\mathbb{R},\ x>0,\ y>0,\ x+y=1\} $$

Prove that:



(i) $\frac{2ab}{a+b}\in M;$



(ii) $\sqrt{ab}\in M.$
2001 Romania National Olympiad
We consider a right trapezoid $ABCD$, in which $AB||CD,AB>CD,AD\perp AB$ and $AD>CD$. The diagonals $AC$ and $BD$ intersect at $O$. The parallel through $O$ to $AB$ intersects $AD$ in $E$ and $BE$ intersects $CD$ in $F$. Prove that $CE\perp AF$ if and only if $AB\cdot CD=AD^2-CD^2$ .
2001 Romania National Olympiad
Consider the acute angle $ABC$. On the half-line $BC$ we consider the distinct points $P$ and $Q$ whose projections onto the line $AB$ are the points $M$ and $N$. Knowing that $AP=AQ$ and $AM^2-AN^2=BN^2-BM^2$, find the angle $ABC$.
2001 Romania National Olympiad
Determine all real numbers $a$ and $b$ such that $a+b\in\mathbb{Z}$ and $a^2+b^2=2$.
2001 Romania National Olympiad
For every rational number $m>0$ we consider the function $f_m:\mathbb{R}\rightarrow\mathbb{R},f_m(x)=\frac{1}{m}x+m$. Denote by $G_m$ the graph of the function $f_m$. Let $p,q,r$ be positive rational numbers.



a) Show that if $p$ and $q$ are distinct then $G_p\cap G_q$ is non-empty.



b) Show that if $G_p\cap G_q$ is a point with integer coordinates, then $p$ and $q$ are integer numbers.



c) Show that if $p,q,r$ are consecutive natural numbers, then the area of the triangle determined by intersections of $G_p,G_q$ and $G_r$ is equal to $1$.
2001 Romania National Olympiad
We consider the points $A,B,C,D$, not in the same plane, such that $AB\perp CD$ and $AB^2+CD^2=AD^2+BC^2$.



a) Prove that $AC\perp BD$.



b) Prove that if $CD<BC<BD$, then the angle between the planes $(ABC)$ and $(ADC)$ is greater than $60^{\circ}$.
2001 Romania National Olympiad
In the cube $ABCDA'B'C'D'$, with side $a$, the plane $(AB'D')$ intersects the planes $(A'BC),(A'CD),(A'DB)$ after the lines $d_1,d_2$ and $d_3$ respectively.



a) Show that the lines $d_1,d_2,d_3$ intersect pairwise.



b) Determine the area of the triangle formed by these three lines.
2001 Romania National Olympiad
Let $A$ be a set of real numbers which verifies:

$$ a)\ 1 \in A \\ b) \ x\in A\implies x^2\in A\\ c)\ x^2-4x+4\in A\implies x\in A $$

Show that $2000+\sqrt{2001}\in A$.
2001 Romania National Olympiad
Let $ABC$ be a triangle $(A=90^{\circ})$ and $D\in (AC)$ such that $BD$ is the bisector of $B$. Prove that $BC-BD=2AB$ if and only if

$$\frac{1}{BD}-\frac{1}{BC}=\frac{1}{2AB} $$
2001 Romania National Olympiad
Let $n\in\mathbb{N}^*$ and $v_1,v_2,\ldots ,v_n$ be vectors in the plane with lengths less than or equal to $1$. Prove that there exists $\xi_1,\xi_2,\ldots ,\xi_n\in\{-1,1\}$ such that

$$ | \xi_1v_1+\xi_2v_2+\ldots +\xi_nv_n|\le\sqrt{2}$$
2001 Romania National Olympiad
Determine the ordered systems $(x,y,z)$ of positive rational numbers for which $x+\frac{1}{y},y+\frac{1}{z}$ and $z+\frac{1}{x}$ are integers.
2001 Romania National Olympiad
Let $a$ and $b$ be complex non-zero numbers and $z_1,z_2$ the roots of the polynomials $X^2+aX+b$. Show that $|z_1+z_2|=|z_1|+|z_2|$ if and only if there exists a real number $\lambda\ge 4$ such that $a^2=\lambda b$.
2001 Romania National Olympiad
In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality

$$\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma $$
2001 Romania National Olympiad
Let $m,k$ be positive integers, $k<m$ and $M$ a set with $m$ elements. Prove that the maximal number of subsets $A_1,A_2,\ldots ,A_p$ of $M$ for which $A_i\cap A_j$ has at most $k$ elements, for every $1\le i<j\le p$, equals

$$ p_{max}=\binom{m}{0}+\binom{m}{1}+\binom{m}{2}+\ldots+\binom{m}{k+1}$$
2001 Romania National Olympiad
Let $n\ge 2$ be an even integer and $a,b$ real numbers such that $b^n=3a+1$. Show that the polynomial $P(X)=(X^2+X+1)^n-X^n-a$ is divisible by $Q(X)=X^3+X^2+X+b$ if and only if $b=1$.
2001 Romania National Olympiad
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, derivable on $R\backslash\{x_0\}$, having finite side derivatives in $x_0$. Show that there exists a derivable function $g:\mathbb{R}\rightarrow\mathbb{R}$, a linear function $h:\mathbb{R}\rightarrow\mathbb{R}$ and $\alpha\in\{-1,0,1\}$ such that:

$$ f(x)=g(x)+\alpha |h(x)|,\ \forall x\in\mathbb{R} $$
2001 Romania National Olympiad
We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$.



a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command.

B=%Error. "rank" is a bad command.

C = r$, such that $A=BC$.



b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.