Navvye/Olympiad-Problems-CSV huggingface dataset:
2007 Romania Team Selection Test
Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company,  while for $n < 11$ this is not anymore necessarily true.



Dan Schwarz
2007 Romania Team Selection Test
For $n\in\mathbb{N}$, $n\geq 2$, $a_{i}, b_{i}\in\mathbb{R}$, $1\leq i\leq n$, such that $$\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0. $$ Prove that

$$\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n. $$

Cezar Lupu & Tudorel Lupu
2007 Romania Team Selection Test
Let $ ABC$ be a triangle, let $ E, F$ be the tangency points of the incircle $ \Gamma(I)$ to the sides $ AC$, respectively $ AB$, and let $ M$ be the midpoint of the side $ BC$. Let $ N = AM \cap EF$, let $ \gamma(M)$ be the circle of diameter $ BC$, and let $ X, Y$ be the other (than $ B, C$) intersection points of $ BI$, respectively $ CI$, with $ \gamma$. Prove that

$$ \frac {NX} {NY} = \frac {AC} {AB}.

$$

Cosmin Pohoata
2007 Romania Team Selection Test
The problem is about real polynomial functions, denoted by $f$, of degree $\deg f$.



a) Prove that a polynomial function $f$ can`t be wrriten as sum of at most $\deg f$ periodic functions.



b) Show that if a polynomial function of degree $1$ is written as sum of two periodic functions, then they are unbounded on every interval (thus, they are "wild").



c) Show that every polynomial function of degree $1$ can be written as sum of two periodic functions.



d) Show that every polynomial function $f$ can be written as sum of $\deg f+1$ periodic functions.



e) Give an example of a function that can`t be written as a finite sum of periodic functions.



Dan Schwarz
2006 Romania Team Selection Test
Let  $ABC$  and $AMN$ be two similar triangles with the same orientation, such that  $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral.



Valentin Vornicu
2006 Romania Team Selection Test
Let  $p$ a prime number, $p\geq 5$. Find the number of polynomials of the form

$$ x^p + px^k + p x^l + 1, \quad k > l, \quad k, l \in \left\{1,2,\cdots,p-1\right\},  $$ which are irreducible in $\mathbb{Z}[X]$.



Valentin Vornicu
2006 Romania Team Selection Test
The real numbers  $a_1,a_2,\dots,a_n$ are given such that $|a_i|\leq 1$

for all  $i=1,2,\dots,n$ and

$a_1+a_2+\cdots+a_n=0$.



a) Prove that there exists $k\in\{1,2,\dots,n\}$ such that

$$ |a_1+2a_2+\cdots+ka_k|\leq\frac{2k+1}{4}.  $$



b) Prove that for  $n > 2$ the bound above is the best possible.



Radu Gologan, Dan Schwarz
2006 Romania Team Selection Test
Let $\{a_n\}_{n\geq 1}$ be a sequence with $a_1=1$, $a_2=4$ and for all $n>1$, $$ a_{n} = \sqrt{ a_{n-1}a_{n+1} + 1 } . $$

a) Prove that all the terms of the sequence are positive integers.



b) Prove that $2a_na_{n+1}+1$ is a perfect square for all positive integers $n$.



Valentin Vornicu
2006 Romania Team Selection Test
Let $ABC$ be a triangle with $\angle B = 30^{\circ }$. We consider the closed disks of radius $\frac{AC}3$, centered in $A$, $B$, $C$. Does there exist an equilateral triangle with one vertex in each of the 3 disks?



Radu Gologan, Dan Schwarz
2006 Romania Team Selection Test
For which pairs of positive integers $(m,n)$ there exists a set $A$ such that for all positive integers $x,y$, if $|x-y|=m$, then at least one of the numbers  $x,y$ belongs to the set $A$, and if $|x-y|=n$, then at least one of the numbers $x,y$ does not belong to the set $A$?



Adapted by Dan Schwarz from A.M.M.
2006 Romania Team Selection Test
Let $x_i$, $1\leq i\leq n$ be real numbers. Prove that $$ \sum_{1\leq i<j\leq n}|x_i+x_j|\geq\frac{n-2}{2}\sum_{i=1}^n|x_i|. $$

Discrete version by Dan Schwarz of a Putnam problem
2006 Romania Team Selection Test
The circle of center $I$ is inscribed in the convex quadrilateral $ABCD$. Let $M$ and $N$ be points on the segments $AI$ and $CI$, respectively, such that $\angle MBN = \frac 12 \angle ABC$. Prove that $\angle MDN = \frac 12 \angle ADC$.
2006 Romania Team Selection Test
Let $A$ be point in the exterior of the circle $\mathcal C$. Two lines passing through $A$ intersect the circle $\mathcal C$ in points $B$ and $C$ (with $B$ between $A$ and $C$) respectively in $D$ and $E$ (with $D$ between $A$ and $E$). The parallel from $D$ to $BC$ intersects the second time the circle $\mathcal C$ in $F$. Let $G$ be the second point of intersection between the circle $\mathcal C$ and the line $AF$ and $M$ the point in which the lines $AB$ and $EG$ intersect. Prove that

$$ \frac 1{AM} = \frac 1{AB} + \frac 1{AC}. $$
2006 Romania Team Selection Test
Let $\gamma$ be the incircle in the triangle $A_0A_1A_2$. For all $i\in\{0,1,2\}$ we make the following constructions (all indices are considered modulo 3): $\gamma_i$ is the circle tangent to $\gamma$ which passes through the points $A_{i+1}$ and $A_{i+2}$; $T_i$ is the point of tangency between $\gamma_i$ and $\gamma$; finally, the common tangent in $T_i$ of $\gamma_i$ and $\gamma$ intersects the line $A_{i+1}A_{i+2}$ in the point $P_i$. Prove that



a) the points $P_0$, $P_1$ and $P_2$ are collinear;



b) the lines $A_0T_0$, $A_1T_1$ and $A_2T_2$ are concurrent.
2006 Romania Team Selection Test
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that: $$ \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2. $$
2006 Romania Team Selection Test
Let $r$ and $s$ be two rational numbers. Find all functions $f: \mathbb Q \to \mathbb Q$ such that for all $x,y\in\mathbb Q$ we have $$ f(x+f(y)) = f(x+r)+y+s. $$
2006 Romania Team Selection Test
Find all non-negative integers $m,n,p,q$ such that $$ p^mq^n = (p+q)^2 +1 . $$
2006 Romania Team Selection Test
Let $n>1$ be an integer. A set $S \subset \{ 0,1,2, \ldots, 4n-1\}$ is called rare if, for any $k\in\{0,1,\ldots,n-1\}$, the following two conditions take place at the same time



(1) the set $S\cap \{4k-2,4k-1,4k, 4k+1, 4k+2 \}$ has at most two elements;



(2) the set $S\cap \{4k+1,4k+2,4k+3\}$ has at most one element.



Prove that the set $\{0,1,2,\ldots,4n-1\}$ has exactly $8 \cdot 7^{n-1}$ rare subsets.
2006 Romania Team Selection Test
Let $p$, $q$ be two integers, $q\geq p\geq 0$. Let $n \geq 2$ be an integer and $a_0=0, a_1 \geq 0, a_2, \ldots, a_{n-1},a_n = 1$ be real numbers such that $$ a_{k} \leq \frac{ a_{k-1} + a_{k+1} } 2 , \ \forall \ k=1,2,\ldots, n-1 . $$ Prove that $$ (p+1) \sum_{k=1}^{n-1} a_k^p \geq (q+1) \sum_{k=1}^{n-1} a_k^q . $$
2006 Romania Team Selection Test
Let $n$ be a positive integer of the form $4k+1$, $k\in \mathbb N$ and $A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}$. Prove that there exist integers $x,y$ such that $x^n+y^n \in A$ and $x+y \notin A$.
2006 Romania Team Selection Test
Let $m$ and $n$ be positive integers and $S$ be a subset with $(2^m-1)n+1$ elements of the set $\{1,2,3,\ldots, 2^mn\}$. Prove that $S$ contains $m+1$ distinct numbers $a_0,a_1,\ldots, a_m$ such that $a_{k-1} \mid a_{k}$ for all $k=1,2,\ldots, m$.
2006 Romania Team Selection Test
Let $x_1=1$, $x_2$, $x_3$, $\ldots$ be a sequence of real numbers such that for all $n\geq 1$ we have $$ x_{n+1} = x_n + \frac 1{2x_n} . $$ Prove that $$ \lfloor 25 x_{625} \rfloor = 625 . $$
2006 Romania Team Selection Test
Let $ABC$ be an acute triangle with $AB \neq AC$. Let $D$ be the foot of the altitude from $A$ and $\omega$ the circumcircle of the triangle. Let $\omega_1$ be the circle tangent to $AD$, $BD$ and $\omega$. Let $\omega_2$ be the circle tangent to $AD$, $CD$ and $\omega$. Let $\ell$ be the interior common tangent to both $\omega_1$ and $\omega_2$, different from $AD$.



Prove that $\ell$ passes through the midpoint of $BC$ if and only if $2BC = AB + AC$.
2005 Romania Team Selection Test
Solve the equation $3^x=2^xy+1$ in positive integers.
2005 Romania Team Selection Test
Let $n\geq 1$ be an integer and let $X$ be a set of $n^2+1$ positive integers such that in any subset of $X$ with $n+1$ elements there exist two elements $x\neq y$ such that $x\mid y$. Prove that there exists a subset $\{x_1,x_2,\ldots, x_{n+1} \} \in X$ such that  $x_i \mid x_{i+1}$ for all $i=1,2,\ldots, n$.
2005 Romania Team Selection Test
Prove that if the distance from a point inside a convex polyhedra with $n$ faces to the vertices of the polyhedra is at most 1, then the sum of the distances from this point to the faces of the polyhedra is smaller than $n-2$.



Calin Popescu
2005 Romania Team Selection Test
Prove that in any convex polygon with $4n+2$ sides ($n\geq 1$) there exist two consecutive sides which form a triangle of area at most $\frac 1{6n}$ of the area of the polygon.
2005 Romania Team Selection Test
Let $m,n$ be co-prime integers, such that $m$ is even and $n$ is odd. Prove that the following expression does not depend on the values of $m$ and $n$:

$$ \frac 1{2n} + \sum^{n-1}_{k=1} (-1)^{\left[ \frac{mk}n \right]} \left\{ \frac {mk}n \right\} . $$



Bogdan Enescu
2005 Romania Team Selection Test
A sequence of real numbers $\{a_n\}_n$ is called a bs sequence if $a_n = |a_{n+1} - a_{n+2}|$, for all $n\geq 0$. Prove that a bs sequence is bounded if and only if the function $f$ given by $f(n,k)=a_na_k(a_n-a_k)$, for all $n,k\geq 0$  is the null function.



Mihai Baluna - ISL 2004
2005 Romania Team Selection Test
Let $A_0A_1A_2A_3A_4A_5$ be a convex hexagon inscribed in a circle. Define the points $A_0'$, $A_2'$, $A_4'$ on the circle, such that

$$ A_0A_0' \parallel A_2A_4, \quad A_2A_2' \parallel A_4A_0, \quad A_4A_4' \parallel A_2A_0 . $$



Let the lines $A_0'A_3$ and $A_2A_4$ intersect in $A_3'$, the lines $A_2'A_5$ and $A_0A_4$ intersect in $A_5'$ and the lines $A_4'A_1$ and $A_0A_2$ intersect in $A_1'$.



Prove that if the lines $A_0A_3$, $A_1A_4$ and $A_2A_5$ are concurrent then the lines $A_0A_3'$, $A_4A_1'$ and $A_2A_5'$ are also concurrent.
2005 Romania Team Selection Test
Let $ABC$ be a triangle, and let $D$, $E$, $F$ be 3  points on the sides $BC$, $CA$ and $AB$ respectively, such that the inradii of the triangles $AEF$, $BDF$ and $CDE$ are equal with half of the inradius of the triangle $ABC$. Prove that $D$, $E$, $F$ are the midpoints of the sides of the triangle $ABC$.
2005 Romania Team Selection Test
Let $P$ be a polygon (not necessarily convex) with $n$ vertices, such that all its sides and diagonals are less or equal with 1 in length. Prove that the area of the polygon is less than $\dfrac {\sqrt 3} 2$.
2005 Romania Team Selection Test
Let $a\in\mathbb{R}-\{0\}$. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$.



Dan Schwartz
2005 Romania Team Selection Test
On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out.

Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit.



Dan Schwartz
2005 Romania Team Selection Test
Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that

$$ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . $$



Călin Popescu
2005 Romania Team Selection Test
a) Prove that there exists a sequence of digits $\{c_n\}_{n\geq 1}$ such that or each $n\geq 1$ no matter how we interlace $k_n$ digits, $1\leq k_n\leq 9$, between $c_n$ and $c_{n+1}$, the infinite sequence thus obtained does not represent the fractional part of a rational number.



b) Prove that for $1\leq k_n\leq 10$ there is no such sequence $\{c_n\}_{n\geq 1}$.



Dan Schwartz
2005 Romania Team Selection Test
On a $2004 \times 2004$ chess table there are 2004 queens such that no two are attacking each other\footnote[1]{two queens attack each other if they lie on the same row, column or direction parallel with on of the main diagonals of the table}.



Prove that there exist two queens such that in the rectangle in which the center of the squares on which the queens lie are two opposite corners, has a semiperimeter of 2004.
2005 Romania Team Selection Test
Let $n\geq 2$ be an integer. Find the smallest real value $\rho (n)$ such that for any $x_i>0$, $i=1,2,\ldots,n$ with $x_1 x_2 \cdots x_n = 1$, the inequality

$$ \sum_{i=1}^n \frac 1{x_i} \leq \sum_{i=1}^n x_i^r $$ is true for all $r\geq \rho (n)$.
2005 Romania Team Selection Test
Let $\mathbb{N}=\{1,2,\ldots\}$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is  a divisor of $(m^2+n)^2$.
2005 Romania Team Selection Test
We consider a polyhedra which has exactly two vertices adjacent with an odd number of edges, and these two vertices are lying on the same edge.



Prove that for all integers $n\geq 3$ there exists a face of the polyhedra with a number of sides not divisible by $n$.
2004 Romania Team Selection Test
Let $a_1,a_2,a_3,a_4$ be the sides of an arbitrary quadrilateral of perimeter $2s$. Prove that

$$ \sum\limits^4_{i=1} \dfrac 1{a_i+s} \leq \dfrac 29\sum\limits_{1\leq i<j\leq 4} \dfrac 1{ \sqrt { (s-a_i)(s-a_j)}}.  $$

When does the equality hold?
2004 Romania Team Selection Test
Let $\{R_i\}_{1\leq i\leq n}$ be a family of disjoint closed rectangular surfaces with total area 4 such that their projections of the $Ox$ axis is an interval. Prove that there exist a triangle with vertices in $\displaystyle \bigcup_{i=1}^n R_i$ which has an area of at least 1.



[Thanks Grobber for the correction]
2004 Romania Team Selection Test
Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds:

$$ f(f(n)) \leq \frac {n+f(n)} 2 . $$
2004 Romania Team Selection Test
Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$  there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.
2004 Romania Team Selection Test
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where $$ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . $$
2004 Romania Team Selection Test
Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$, $x_1=0$, $x_2=2c$, $x_3=3b$ and for all positive integers $n$ we have

$$ x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} . $$

Prove that for all positive integers $m$, and for all primes $p$ the number $x_{p^m}$ is divisible by $p$.
2004 Romania Team Selection Test
Let $\Gamma$ be a circle, and let $ABCD$ be a square lying inside the circle $\Gamma$. Let $\mathcal{C}_a$ be a circle tangent interiorly  to $\Gamma$, and also tangent to the sides $AB$ and $AD$ of the square, and also lying inside the opposite angle of $\angle BAD$. Let $A'$ be the tangency point of the two circles. Define similarly the circles $\mathcal{C}_b$, $\mathcal{C}_c$, $\mathcal{C}_d$ and the points $B',C',D'$ respectively.



Prove that the lines $AA'$, $BB'$, $CC'$ and $DD'$ are concurrent.
2004 Romania Team Selection Test
Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements.



Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.
2004 Romania Team Selection Test
Prove that for all positive integers $n,m$, with $m$ odd, the following number is an integer

$$ \frac 1{3^mn}\sum^m_{k=0} { 3m \choose 3k } (3n-1)^k. $$
2004 Romania Team Selection Test
Let $I$ be the incenter of the non-isosceles triangle $ABC$ and let $A',B',C'$ be the tangency points of the incircle with the sides $BC,CA,AB$ respectively. The lines $AA'$ and $BB'$ intersect in $P$, the lines $AC$ and $A'C'$ in $M$ and the lines $B'C'$ and $BC$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular.



Alternative formulation. The incircle of a non-isosceles triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$ and $AB$ in $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, respectively. The lines $AA^{\prime}$ and $BB^{\prime}$ intersect in $P$, the lines $AC$ and $A^{\prime}C^{\prime}$ intersect in $M$, and the lines $BC$ and $B^{\prime}C^{\prime}$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular.
2004 Romania Team Selection Test
Let $n\geq 2$ be an integer and let $a_1,a_2,\ldots,a_n$ be real numbers. Prove that for any non-empty subset $S\subset \{1,2,3,\ldots, n\}$ we have

$$ \left( \sum_{i \in S} a_i \right)^2 \leq \sum_{1\leq i \leq j \leq n } (a_i + \cdots + a_j ) ^2 .  $$

Gabriel Dospinescu
2004 Romania Team Selection Test
Let $m\geq 2$ be an integer. A positive integer $n$ has the property that for any positive integer $a$ coprime with $n$, we have $a^m - 1\equiv 0 \pmod n$.



Prove that $n \leq 4m(2^m-1)$.



Created by Harazi, modified by Marian Andronache.
2004 Romania Team Selection Test
Let $O$ be a point in the plane of the triangle $ABC$. A circle $\mathcal{C}$ which passes through $O$ intersects the second time the lines $OA,OB,OC$ in $P,Q,R$ respectively. The circle $\mathcal{C}$ also intersects for the second time the circumcircles of the triangles $BOC$, $COA$ and $AOB$ respectively in $K,L,M$.



Prove that the lines $PK,QL$ and $RM$ are concurrent.
2004 Romania Team Selection Test
Some of the $n$ faces of a polyhedron are colored in black such that any two black-colored faces have no common vertex. The rest of the faces of the polyhedron are colored in white.



Prove that the number of common sides of two white-colored faces of the polyhedron is at least $n-2$.
2004 Romania Team Selection Test
Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$.



Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that

$$ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. $$
2004 Romania Team Selection Test
On a chess table $n\times m$ we call a move  the following succesion of operations



(i) choosing some unmarked squares, any two not lying on the same row or column;

(ii) marking them with 1;

(iii) marking with 0 all the unmarked squares which lie on the same line and column with a square marked with the number 1 (even if the square has been marked with 1 on another move).



We call a game a succession of moves that end in the moment that we cannot make any more moves.



What is the maximum possible sum of the numbers on the table at the end of a game?
2004 Romania Team Selection Test
Let $p$ be a prime number and $f\in \mathbb{Z}[X]$ given by

$$ f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 , $$where $a_i = \left( \tfrac ip\right)$ is the Legendre symbol of $i$ with respect to $p$ (i.e. $a_i=1$ if $ i^{\frac {p-1}2} \equiv 1 \pmod p$ and $a_i=-1$ otherwise, for all $i=1,2,\ldots,p-1$).



a) Prove that $f(x)$ is divisible with $(x-1)$, but not with $(x-1)^2$ iff $p \equiv 3 \pmod 4$;

b) Prove that if $p\equiv 5 \pmod 8$ then $f(x)$ is divisible with $(x-1)^2$ but not with $(x-1)^3$.



Sugested by Calin Popescu.
2003 Romania Team Selection Test
Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$

$$ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}.  $$

Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.
2003 Romania Team Selection Test
Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Consider a point $P$ inside the triangle having $PA=1$, $PB=2$ and $PC=3$. Find the maximum possible area of the triangle $ABC$.
2003 Romania Team Selection Test
Let $n,k$ be positive integers such that $n^k>(k+1)!$ and consider the set

$$ M=\{(x_1,x_2,\ldots,x_n)\dvd x_i\in\{1,2,\ldots,n\},\ i=\overline{1,k}\}. $$

Prove that if $A\subset M$ has $(k+1)!+1$ elements, then there are two elements $\{\alpha,\beta\}\subset A$, $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)$, $\beta=(\beta_1,\beta_2,\ldots,\beta_n)$ such that

$$ (k+1)! \left| (\beta_1-\alpha_1)(\beta_2-\alpha_2)\cdots (\beta_k-\alpha_k) \right. .$$
2003 Romania Team Selection Test
Prove that among the elements of the sequence $\left\{ \left\lfloor n\sqrt{2003} \right\rfloor \right\}_{n\geq 1}$ one can find a geometric progression having any number of terms, and having the ratio bigger than $k$, where $k$ can  be any positive integer.



Radu Gologan
2003 Romania Team Selection Test
Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring  of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials.



Mihai Piticari
2003 Romania Team Selection Test
At a math contest there are $2n$ students participating. Each of them submits a problem to the jury, which thereafter gives each students one of the $2n$ problems submitted. One says that the contest is fair is there are $n$ participants which receive their problems from the other $n$ participants.



Prove that the number of distributions of the problems in order to obtain a fair contest is a perfect square.
2003 Romania Team Selection Test
Find all integers $a,b,m,n$, with $m>n>1$, for which the polynomial $f(X)=X^n+aX+b$ divides the polynomial $g(X)=X^m+aX+b$.



Laurentiu Panaitopol
2003 Romania Team Selection Test
Two circles $\omega_1$ and $\omega_2$ with radii $r_1$ and $r_2$, $r_2>r_1$, are externally tangent. The  line $t_1$ is tangent to the circles $\omega_1$ and $\omega_2$ at points $A$ and $D$ respectively. The parallel  line $t_2$ to the line $t_1$ is tangent to the circle $\omega_1$ and intersects the circle $\omega_2$ at points $E$ and $F$. The line $t_3$ passing through $D$ intersects the line $t_2$ and the circle $\omega_2$ in $B$ and $C$ respectively, both different of $E$ and $F$ respectively. Prove that the circumcircle of the triangle $ABC$ is tangent to the line $t_1$.



Dinu Serbanescu
2003 Romania Team Selection Test
Let $n\geq 3$ be a positive integer. Inside a $n\times n$ array there are placed $n^2$ positive numbers with sum $n^3$. Prove that we can find a square $2\times 2$ of 4 elements of the array, having the sides parallel with the sides of the array, and for which the sum of the elements in the square is greater than $3n$.



Radu Gologan
2003 Romania Team Selection Test
Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$.



Valentin Vornicu
2003 Romania Team Selection Test
In a square of side 6 the points $A,B,C,D$ are given such that the distance between any two of the four points is at least 5. Prove that $A,B,C,D$ form a convex quadrilateral and its area is greater than 21.



Laurentiu Panaitopol
2003 Romania Team Selection Test
A word is a sequence of n letters of the alphabet {a, b, c, d}. A word is said to be complicated if it contains two consecutive groups of identic letters. The words caab, baba and cababdc, for example, are complicated words, while bacba and dcbdc are not. A word that is not complicated is a simple word. Prove that the numbers of simple words with n letters is greater than $2^n$, if n is a positive integer.
2003 Romania Team Selection Test
A parliament has $n$ senators. The senators form 10 parties and 10 committees, such that any senator belongs to exactly one party and one committee. Find the least possible $n$ for which it is possible to label the parties and the committees with numbers from 1 to 10, such that there are at least 11 senators for which the numbers of the corresponding party and committee are equal.