Navvye/Olympiad-Problems-CSV huggingface dataset:
2007 Romania National Olympiad
Given a set $A$ and a function $f: A\rightarrow A$, denote by $f_{1}(A)=f(A)$, $f_{2}(A)=f(f_{1}(A))$, $f_{3}(A)=f(f_{2}(A))$, and so on, ($f_{n}(A)=f(f_{n-1}(A))$, where the notation $f(B)$ means the set $\{ f(x) \ : \ x\in B\}$ of images of points from $B$).

Denote also by $f_{\infty}(A)=f_{1}(A)\cap f_{2}(A)\cap \ldots = \bigcap_{n\geq 1}f_{n}(A)$.



a) Show that if $A$ is finite, then $f(f_{\infty}(A))=f_{\infty}(A)$.



b) Determine if the above is true for $A=\mathbb{N}\times \mathbb{N}$ and the function

$$f\big((m,n)\big)=\begin{cases}(m+1,n) & \mbox{if }n\geq m\geq 1 \\ (0,0) & \mbox{if }m>n \\ (0,n+1) & \mbox{if }n=0. \end{cases}$$
2007 Romania National Olympiad
Show that the equation $z^{n}+z+1=0$ has a solution with $|z|=1$ if and only if $n-2$ is divisble by $3$.
2007 Romania National Olympiad
Solve the equation $$2^{x^{2}+x}+\log_{2}x = 2^{x+1}$$
2007 Romania National Olympiad
For which integers $n\geq 2$, the number $(n-1)^{n^{n+1}}+(n+1)^{n^{n-1}}$ is divisible by $n^{n}$ ?
2007 Romania National Olympiad
a) For a finite set of natural numbers $S$, denote by $S+S$ the set of numbers $z=x+y$, where $x,y\in S$. Let $m=|S|$. Show that $|S+S|\leq \frac{m(m+1)}{2}$.



b) Let $m$ be a fixed positive integer. Denote by $C(m)$ the greatest integer $k\geq 1$ for which there exists a set $S$ of $m$ integers, such that $\{1,2,\ldots,k\}\subseteq S\cup(S+S)$. For example, $C(3)=8$, with $S=\{1,3,4\}$. Show that $\frac{m(m+6)}{4}\leq C(m) \leq \frac{m(m+3)}{2}$.
2007 Romania National Olympiad
In a triangle $ ABC$, where $ a = BC$, $ b = CA$ and $ c = AB$, it is known that: $ a + b - c = 2$ and $ 2ab - c^2 = 4$. Prove that $ ABC$ is an equilateral triangle.
2007 Romania National Olympiad
Consider the triangle $ ABC$ with $ m(\angle BAC = 90^\circ)$ and $ AC = 2AB$. Let $ P$ and $ Q$ be the midpoints of $ AB$ and $ AC$,respectively. Let $ M$ and $ N$ be two points found on the side $ BC$ such that $ CM = BN = x$. It is also known that $ 2S[MNPQ] = S[ABC]$. Determine $ x$ in function of $ AB$.
2007 Romania National Olympiad
Consider the triangle $ ABC$ with $ m(\angle BAC) = 90^\circ$ and $ AB < AC$.Let a point $ D$ on the side $ AC$ such that: $ m(\angle ACB) = m(\angle DBA)$.Let $ E$ be a point on the side $ BC$ such that $ DE\perp BC$.It is known that $ BD + DE = AC$. Find the measures of the angles in the triangle $ ABC$.
2007 Romania National Olympiad
Let $ m,n$ be two natural numbers with $ m > 1$ and $ 2^{2m + 1} - n^2\geq 0$. Prove that:

$$ 2^{2m + 1} - n^2\geq 7 .$$
2007 Romania National Olympiad
Prove that the number $ 10^{10}$ can't be written as the product of two natural numbers which do not contain the digit "$ 0$" in their decimal representation.
2007 Romania National Olympiad
In a building there are 6018 desks in 2007 rooms, and in every room there is at least one desk. Every room can be cleared dividing the desks in the oher rooms such that in every room is the same number of desks. Find out what methods can be used for dividing the desks initially.
2007 Romania National Olympiad
a) In a triangle $ MNP$, the lenghts of the sides are less than $ 2$. Prove that the lenght of the altitude corresponding to the side $ MN$ is less than $ \sqrt {4 - \frac {MN^2}{4}}$.



b) In a tetrahedron $ ABCD$, at least $ 5$ edges have their lenghts less than $ 2$.Prove that the volume of the tetrahedron is less than $ 1$.
2007 Romania National Olympiad
Let $ ABCD$ be a tetrahedron.Prove that if a point $ M$ in a space satisfies the relation:

\begin{align*} MA^2 + MB^2 + CD^2 &= MB^2 + MC^2 + DA^2 \\ &= MC^2 + MD^2 + AB^2 \\ &= MD^2 + MA^2 + BC^2 . \end{align*}

then it is found on the common perpendicular of the lines $ AC$ and $ BD$.
2006 Romania National Olympiad
Let $ABC$ be a triangle and the points $M$ and $N$ on the sides $AB$ respectively $BC$, such that $2 \cdot \frac{CN}{BC} = \frac{AM}{AB}$. Let $P$ be a point on the line $AC$. Prove that the lines $MN$ and $NP$ are perpendicular if and only if $PN$ is the interior angle bisector of $\angle MPC$.
2006 Romania National Olympiad
A square of side $n$ is formed from $n^2$ unit squares, each colored in red, yellow or green. Find minimal $n$, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).
2006 Romania National Olympiad
In the acute-angle triangle $ABC$ we have $\angle ACB = 45^\circ$. The points $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$, and $H$ is the orthocenter of the triangle. We consider the points $D$ and $E$ on the segments $AA_1$ and $BC$ such that $A_1D = A_1E = A_1B_1$. Prove that



a) $A_1B_1 = \sqrt{ \frac{A_1B^2+A_1C^2}{2} }$;



b) $CH=DE$.
2006 Romania National Olympiad
Let $A$ be a set of positive integers with at least 2 elements. It is given that for any numbers $a>b$, $a,b \in A$ we have $\frac{ [a,b] }{ a- b } \in A$, where by $[a,b]$ we have denoted the least common multiple of $a$ and $b$. Prove that the set $A$ has exactly two elements.



Marius Gherghu, Slatina
2006 Romania National Olympiad
We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.
2006 Romania National Olympiad
Let $n$ be a positive integer. Prove that there exists an integer $k$, $k\geq 2$, and numbers $a_i \in \{ -1, 1 \}$, such that $$ n = \sum_{1\leq i < j \leq k } a_ia_j . $$
2006 Romania National Olympiad
Let $ABCDA_1B_1C_1D_1$ be a cube and $P$ a variable point on the side $[AB]$. The perpendicular plane on $AB$ which passes through $P$ intersects the line $AC'$ in $Q$. Let $M$ and $N$ be the midpoints of the segments $A'P$ and $BQ$ respectively.



a) Prove that the lines $MN$ and $BC'$ are perpendicular if and only if $P$ is the midpoint of $AB$.



b) Find the minimal value of the angle between the lines $MN$ and $BC'$.
2006 Romania National Olympiad
Let $a,b,c \in \left[ \frac 12, 1 \right]$. Prove that $$ 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 .  $$

selected by Mircea Lascu
2006 Romania National Olympiad
Find the maximal value of $$ \left( x^3+1 \right) \left( y^3 + 1\right) ,  $$ where $x,y \in \mathbb R$, $x+y=1$.



Dan Schwarz
2006 Romania National Olympiad
Let $\displaystyle ABC$ and $\displaystyle DBC$ be isosceles triangle with the base $\displaystyle BC$. We know that $\displaystyle \measuredangle ABD = \frac{\pi}{2}$. Let $\displaystyle M$ be the midpoint of $\displaystyle BC$. The points $\displaystyle E,F,P$ are chosen such that $\displaystyle E \in (AB)$, $\displaystyle P \in (MC)$, $\displaystyle C \in (AF)$, and $\displaystyle \measuredangle BDE = \measuredangle ADP = \measuredangle CDF$. Prove that $\displaystyle P$ is the midpoint of $\displaystyle EF$ and $\displaystyle DP \perp EF$.
2006 Romania National Olympiad
We have a quadrilateral $ABCD$ inscribed in a circle of radius $r$, for which there is a point $P$ on $CD$ such that $CB=BP=PA=AB$.



(a) Prove that there are points $A,B,C,D,P$ which fulfill the above conditions.



(b) Prove that $PD=r$.



Virgil Nicula
2006 Romania National Olympiad
$\displaystyle 2n$ students $\displaystyle (n \geq 5)$ participated at table tennis contest, which took $\displaystyle 4$ days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this:



- there is only one winner;



- there are $\displaystyle 3$ students on the second place;



- no student lost all $\displaystyle 4$ matches.



How many students won only a single match and how many won exactly $\displaystyle 2$ matches? (In the above conditions)
2006 Romania National Olympiad
Let $\displaystyle M$ be a set composed of $\displaystyle n$ elements and let $\displaystyle \mathcal P (M)$ be its power set. Find all functions $\displaystyle f : \mathcal P (M) \to \{ 0,1,2,\ldots,n \}$ that have the properties



(a) $\displaystyle f(A) \neq 0$, for $\displaystyle A \neq \phi$;



(b) $\displaystyle f \left( A \cup B \right) = f \left( A \cap B \right) + f \left( A \Delta B \right)$, for all $\displaystyle A,B \in \mathcal P (M)$, where $\displaystyle A \Delta B = \left( A \cup B \right) \backslash \left( A \cap B \right)$.
2006 Romania National Olympiad
Prove that for all $\displaystyle a,b \in \left( 0 ,\frac{\pi}{4} \right)$ and $\displaystyle n \in \mathbb N^\ast$ we have $$ \frac{\sin^n a + \sin^n b}{\left( \sin a + \sin b \right)^n} \geq \frac{\sin^n 2a + \sin^n 2b}{\left( \sin 2a + \sin 2b \right)^n} . $$
2006 Romania National Olympiad
Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.
2006 Romania National Olympiad
Let $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$. Determine $\displaystyle n$ sets $\displaystyle A_i$, $\displaystyle 1 \leq i \leq n$, from the plane, pairwise disjoint, such that:



(a) for every circle $\displaystyle \mathcal C$ from the plane and for every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$ we have $\displaystyle A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi$;



(b) for all lines $\displaystyle d$ from the plane and every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$, the projection of $\displaystyle A_i$ on $\displaystyle d$ is not $\displaystyle d$.
2006 Romania National Olympiad
Let $A$ be a $n\times n$ matrix with complex elements and let $A^\star$ be the classical adjoint of $A$. Prove that if there exists a positive integer $m$ such that $(A^\star)^m = 0_n$ then $(A^\star)^2 = 0_n$.



Marian Ionescu, Pitesti
2006 Romania National Olympiad
We define a pseudo-inverse $B\in \mathcal M_n(\mathbb C)$  of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations

$$ A = ABA \quad \text{ and } \quad B=BAB. $$

a) Prove that any square matrix has at least a pseudo-inverse.



b) For which matrix $A$ is the pseudo-inverse unique?



Marius Cavachi
2006 Romania National Olympiad
We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$, which have different centers of mass. Prove that there is a point $P$ such that $$ PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n . $$
2006 Romania National Olympiad
Let $f: [0,\infty)\to\mathbb R$ be a function such that for any $x>0$ the sequence $\{f(nx)\}_{n\geq 0}$ is increasing.



a) If the function is also continuous on $[0,1]$ is it true that $f$ is increasing?



b) The same question if the function is continuous on $\mathbb Q \cap [0, \infty)$.
2006 Romania National Olympiad
Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent:



(a) $\displaystyle 1+1=0$;



(b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$, $\displaystyle f \left( X^2 \right)$ is reducible.
2006 Romania National Olympiad
Prove that $$ \lim_{n \to \infty} n \left( \frac{\pi}{4} - n \int_0^1 \frac{x^n}{1+x^{2n}} \, dx \right) = \int_0^1 f(x) \, dx ,  $$ where $f(x) = \frac{\arctan x}{x}$ if $x \in \left( 0,1 \right]$ and $f(0)=1$.



Dorin Andrica, Mihai Piticari
2006 Romania National Olympiad
Let $\displaystyle G$ be a finite group of $\displaystyle n$ elements $\displaystyle ( n \geq 2 )$ and $\displaystyle p$ be the smallest prime factor of $\displaystyle n$. If $\displaystyle G$ has only a subgroup $\displaystyle H$ with $\displaystyle p$ elements, then prove that $\displaystyle H$ is in the center of $\displaystyle G$.



Note. The center of $\displaystyle G$ is the set $\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}$.
2006 Romania National Olympiad
Let $f: [0,1]\to\mathbb{R}$ be a continuous function such that $$ \int_{0}^{1}f(x)dx=0.  $$ Prove that there is $c\in (0,1)$ such that $$ \int_{0}^{c}xf(x)dx=0.  $$

Cezar Lupu, Tudorel Lupu
2005 Romania National Olympiad
Let $ABCD$ be a parallelogram. The interior angle bisector of $\angle ADC$ intersects the line $BC$ in $E$, and the perpendicular bisector of the side $AD$ intersects the line $DE$ in $M$. Let $F= AM \cap BC$. Prove that:



a) $DE=AF$;

b) $AD\cdot AB = DE\cdot DM$.



Daniela and Marius Lobaza, Timisoara
2005 Romania National Olympiad
Let $a,b$ be two integers. Prove that



a) $13 \mid 2a+3b$ if and only if $13 \mid 2b-3a$;

b) If $13 \mid a^2+b^2$ then $13 \mid (2a+3b)(2b+3a)$.



Mircea Fianu
2005 Romania National Olympiad
Let $ABCD$ be a quadrilateral with $AB\parallel CD$ and $AC \perp BD$. Let $O$ be the intersection of $AC$ and $BD$. On the rays $(OA$ and $(OB$ we consider the points $M$ and $N$ respectively such that $\angle ANC = \angle BMD = 90^\circ$. We denote with $E$ the midpoint of the segment $MN$. Prove that



a) $\triangle OMN \sim \triangle OBA$;

b) $OE \perp AB$.



Claudiu-Stefan Popa
2005 Romania National Olympiad
On a circle there are written 2005 non-negative integers with sum 7022. Prove that there exist two pairs formed with two consecutive numbers on the circle such that the sum of the elements in each pair is greater or equal with 8.



After an idea of Marin Chirciu
2005 Romania National Olympiad
We consider a cube with sides of length 1. Prove that a tetrahedron with vertices in the set of the vertices of the cube has the volume $\dfrac 16$ if and only if 3 of the vertices of the tetrahedron are vertices on the same face of the cube.



Dinu Serbanescu
2005 Romania National Olympiad
For a positive integer $n$, written in decimal base, we denote by $p(n)$ the product of its digits.



a) Prove that $p(n) \leq n$;

b) Find all positive integers $n$ such that

$$ 10p(n) = n^2+ 4n - 2005. $$



Eugen Păltănea
2005 Romania National Olympiad
Let the $ABCA'B'C'$ be a regular prism. The points $M$ and $N$ are the midpoints of the sides $BB'$, respectively $BC$, and the angle between the lines $AB'$ and $BC'$ is of $60^\circ$. Let $O$ and $P$ be the intersection of the lines $A'C$ and $AC'$, with respectively $B'C$ and $C'N$.



a) Prove that $AC' \perp (OPM)$;

b) Find the measure of the angle between the line $AP$ and the plane $(OPM)$.



Mircea Fianu
2005 Romania National Olympiad
a) Prove that for all positive reals $u,v,x,y$ the following inequality takes place:

$$ \frac ux + \frac vy \geq \frac {4(uy+vx)}{(x+y)^2} . $$

b) Let $a,b,c,d>0$. Prove that

$$ \frac a{b+2c+d} + \frac b{c+2d+a} + \frac c{d+2a+b} + \frac d{a+2b+c} \geq 1.$$



Traian Tămâian
2005 Romania National Olympiad
Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$. Define the points $E=AD \cap BC$ and $I = AC\cap BD$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$.



Virgil Nicula
2005 Romania National Olympiad
Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which

$$ x(f(x+1)-f(x)) = f(x), $$

for all $x\in\mathbb{R}$ and

$$ | f(x) - f(y) | \leq |x-y| , $$

for all $x,y\in\mathbb{R}$.



Mihai Piticari
2005 Romania National Olympiad
Prove that for all positive integers $n$ there exists a single positive integer divisible with $5^n$ which in decimal base is written using $n$ digits from the set $\{1,2,3,4,5\}$.
2005 Romania National Olympiad
Let $x_1,x_2,\ldots,x_n$ be positive reals. Prove that

$$ \frac 1{1+x_1} + \frac 1{1+x_1+x_2} + \cdots + \frac 1{1+x_1+\cdots + x_n} < \sqrt { \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n}} . $$



Bogdan Enescu
2005 Romania National Olympiad
Let $n$ be a positive integer, $n\geq 2$. For each $t\in \mathbb{R}$, $t\neq k\pi$, $k\in\mathbb{Z}$, we consider the numbers

$$ x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) =  \sum_{k=1}^n k(n-k)\sin{(tk)}. $$

Prove that if $x_n(t) = y_n(t) =0$ if and only if  $\tan {\frac {nt}2} = n \tan {\frac t2}$.



Constantin Buse
2005 Romania National Olympiad
The base $A_{1}A_{2}\ldots A_{n}$ of the pyramid $VA_{1}A_{2}\ldots A_{n}$ is a regular polygon. Prove that if $$\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1},$$ then the pyramid is regular.
2005 Romania National Olympiad
a) Prove that there are no one-to-one (injective) functions $f: \mathbb{N} \to \mathbb{N}\cup \{0\}$ such that

$$ f(mn) = f(m)+f(n) , \ \forall \ m,n \in \mathbb{N}.  $$



b) Prove that for all positive integers $k$ there exist one-to-one functions $f: \{1,2,\ldots,k\}\to\mathbb{N}\cup \{0\}$ such that $f(mn) = f(m)+f(n)$ for all $m,n\in \{1,2,\ldots,k\}$ with $mn\leq k$.



Mihai Baluna
2005 Romania National Olympiad
For $\alpha \in (0,1)$ we consider the equation $\{x\{x\}\}= \alpha$.



a) Prove that the equation has rational solutions if and only if there exist $m,p,q\in\mathbb{Z}$, $0<p<q$, $\gcd(p,q)=1$, such that $\alpha = \left( \frac pq\right)^2 + \frac mq$.

b) Find a solution for $\alpha = \frac {2004}{2005^2}$.
2005 Romania National Olympiad
Let $n\geq 2$ a fixed integer. We shall call a $n\times n$ matrix $A$ with rational elements a radical matrix if there exist an infinity of positive integers $k$, such that the equation $X^k=A$ has solutions in the set of $n\times n$ matrices with rational elements.



a) Prove that if $A$ is a radical matrix then $\det A \in \{-1,0,1\}$ and there exists an infinity of radical matrices with determinant 1;

b) Prove that there exist an infinity of matrices that are not radical and have determinant 0, and also an infinity of matrices that are not radical and have determinant 1.



After an idea of Harazi
2005 Romania National Olympiad
Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function.



a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto;

b) Give an example of such a function.
2005 Romania National Olympiad
Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$, $n\geq 2$. We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$, where $a_{ij}=0$, if $X_i \cap X_j = \emptyset$, and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$.
2005 Romania National Olympiad
Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function.



a) Prove that $f$ is continous;

b) Prove that there exists an unique function $g:[0,\infty)\to\mathbb{R}$ such that for all $x\geq 0$ we have $$ f(x+g(x)) = f(g(x))  - g(x) . $$
2005 Romania National Olympiad
Prove that the group morphisms $f: (\mathbb{C},+)\to(\mathbb{C},+)$ for which there exists a positive $\lambda$ such that $|f(z)| \leq \lambda |z|$ for all $z\in\mathbb{C}$, have the form

$$ f(z) = \alpha z + \beta \overline{z} $$ for some complex $\alpha$, $\beta$.



Cristinel Mortici
2005 Romania National Olympiad
Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\in G$ we denote $H^x = \{ xhx^{-1} \mid h \in H \}$ and we suppose that $H^x \cap H = \{e\}$, for all $x\in G - H$ (where by $e$ we denoted the neutral element of the group $G$).



a) Prove that $H^x=H^y$ if and only if $x^{-1}y \in H$;

b) Find the number of elements of the set $\bigcup_{x\in G} H^x$ as a function of $m$ and $n$.



Calin Popescu
2005 Romania National Olympiad
Let $f:[0,\infty)\to(0,\infty)$ a continous function such that $\lim_{n\to\infty} \int^x_0 f(t)dt$ exists and it is finite. Prove that

$$ \lim_{x\to\infty} \frac 1{\sqrt x} \int^x_0 \sqrt {f(t)} dt = 0. $$



Radu Miculescu
2005 Romania National Olympiad
Let $A$ be a ring with $2^n+1$ elements, where $n$ is a positive integer and let

$$ M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} . $$

Prove that the following statements are equivalent:



a) $A$ is a field;

b) $M$ is not empty and the smallest element in $M$ is $2^n+1$.



Marian Andronache
2004 Romania National Olympiad
On the sides $AB,AD$ of the rhombus $ABCD$ are the points $E,F$ such that $AE=DF$. The lines $BC,DE$ intersect at $P$ and $CD,BF$ intersect at $Q$. Prove that:



(a) $\frac{PE}{PD} + \frac{QF}{QB} = 1$;



(b) $P,A,Q$ are collinear.



Virginia Tica, Vasile Tica
2004 Romania National Olympiad
The sidelengths of a triangle are $a,b,c$.



(a) Prove that there is a triangle which has the sidelengths $\sqrt a,\sqrt b,\sqrt c$.



(b) Prove that $\displaystyle \sqrt{ab}+\sqrt{bc}+\sqrt{ca} \leq a+b+c < 2 \sqrt{ab} + 2 \sqrt{bc} + 2 \sqrt{ca}$.
2004 Romania National Olympiad
Let $ABCD$ be an orthodiagonal trapezoid such that $\measuredangle A = 90^{\circ}$ and $AB$ is the larger base. The diagonals intersect at $O$, $\left( OE \right.$ is the bisector of $\measuredangle AOD$, $E \in \left( AD \right)$ and $EF \| AB$, $F \in \left( BC \right)$. Let $P,Q$ the intersections of the segment $EF$ with $AC,BD$. Prove that:



(a) $EP=QF$;



(b) $EF=AD$.



Claudiu-Stefan Popa
2004 Romania National Olympiad
Let $\mathcal U = \left\{ \left( x,y \right) | x,y \in \mathbb Z, \ 0 \leq x,y < 4 \right\}$.



(a) Prove that we can choose $6$ points from $\mathcal U$ such that there are no isosceles triangles with the vertices among the chosen points.



(b) Prove that no matter how we choose $7$ points from $\mathcal U$, there are always three which form an isosceles triangle.



Radu Gologan, Dinu Serbanescu
2004 Romania National Olympiad
Find all non-negative integers $n$ such that there are $a,b \in \mathbb Z$ satisfying $n^2=a+b$ and $n^3=a^2+b^2$.



Lucian Dragomir
2004 Romania National Olympiad
Prove that the equation $x^2+y^2+z^2+t^2=2^{2004}$, where $0 \leq x \leq y \leq z \leq t$, has exactly $2$ solutions in $\mathbb Z$.



Mihai Baluna
2004 Romania National Olympiad
Let $ABCD A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a truncated regular pyramid in which $BC^{\prime}$ and $DA^{\prime}$ are perpendicular.



(a) Prove that $\measuredangle \left( AB^{\prime},DA^{\prime} \right) = 60^{\circ}$;



(b) If the projection of $B^{\prime}$ on $(ABC)$ is the center of the incircle of $ABC$, then prove that $d \left( CB^{\prime},AD^{\prime} \right) = \frac12 BC^{\prime}$.



Mircea Fianu
2004 Romania National Olympiad
In the interior of a cube of side $6$ there are $1001$ unit cubes with the faces parallel to the faces of the given cube. Prove that there are $2$ unit cubes with the property that the center of one of them lies in the interior or on one of the faces of the other cube.



Dinu Serbanescu
2004 Romania National Olympiad
Find the strictly increasing functions $f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \}$ such that $x+y$ divides $x f(x) + y f(y)$ for all $x,y \in \{ 1,2,\ldots,10 \}$.



Cristinel Mortici
2004 Romania National Olympiad
Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$.



Laurentiu Panaitopol
2004 Romania National Olympiad
Let $H$ be the orthocenter of the acute triangle $ABC$. Let $BB'$ and $CC'$ be altitudes of the triangle ($B^{\prime} \in AC$, $C^{\prime} \in AB$). A variable line $\ell$ passing through $H$ intersects the segments $[BC']$ and $[CB']$ in $M$ and $N$. The perpendicular lines of $\ell$ from $M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$. Determine the locus of the midpoint of the segment $[ PQ]$.



Gheorghe Szolosy
2004 Romania National Olympiad
Let $p,q \in \mathbb N^{\ast}$, $p,q \geq 2$. We say that a set $X$ has the property $\left( \mathcal S \right)$ if no matter how we choose $p$ subsets $B_i \subset X$, $i = \overline{1,n}$, not necessarily distinct, each with $q$ elements, there is a subset $Y \subset X$ with $p$ elements s.t. the intersection of $Y$ with each of the $B_i$'s has an element at most, $i=\overline{1,p}$. Prove that:



(a) if $p=4,q=3$ then any set composed of $9$ elements doesn't have $\left( \mathcal S \right)$;



(b) any set $X$ composed of $pq-q$ elements doesn't have the property $\left( \mathcal S \right)$;



(c) any set $X$ composed of $pq-q+1$ elements has the property $\left( \mathcal S \right)$.



Dan Schwarz