Navvye/Olympiad-Problems-CSV huggingface dataset:
1998 Romania Team Selection Test
The lateral surface of a cylinder of revolution is divided by $n-1$ planes parallel to the base and $m$ parallel generators into $mn$ cases $( n\ge 1,m\ge 3)$. Two cases will be called neighbouring cases if they have a common side. Prove that it is possible to write a real number in each case such that each number is equal to the sum of the numbers of the neighbouring cases and not all the numbers are zero if and only if there exist integers $k,l$ such that $n+1$ does not divide $k$ and

$$ \cos \frac{2l\pi}{m}+\cos\frac{k\pi}{n+1}=\frac{1}{2}$$



Ciprian Manolescu
1997 Romania Team Selection Test
We are given in the plane a line $\ell$ and three circles with centres $A,B,C$ such that they are all tangent to $\ell$ and pairwise externally tangent to each other. Prove that the triangle $ABC$ has an obtuse angle and find all possible values of this this angle.



Mircea Becheanu
1997 Romania Team Selection Test
Find the number of sets $A$ containing $9$ positive integers with the following property: for any positive integer $n\le 500$, there exists a subset $B\subset A$ such that $\sum_{b\in B}{b}=n$.



Bogdan Enescu & Dan Ismailescu
1997 Romania Team Selection Test
Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds:

$$\sum_{P\in\mathcal{C}\cap M} f(P)=0$$

Dorel Mihet
1997 Romania Team Selection Test
Let $ABC$ be a triangle, $D$ be a point on side $BC$, and let $\mathcal{O}$ be the circumcircle of triangle $ABC$. Show that the circles tangent to $\mathcal{O},AD,BD$ and to $\mathcal{O},AD,DC$ are tangent to each other if and only if  $\angle BAD=\angle CAD$.



Dan Branzei
1997 Romania Team Selection Test
Let $VA_1A_2\ldots A_n$ be a pyramid, where $n\ge 4$. A plane $\Pi$ intersects the edges $VA_1,VA_2,\ldots, VA_n$ at the points $B_1,B_2,\ldots,B_n$ respectively such that the polygons $A_1A_2\ldots A_n$ and $B_1B_2\ldots B_n$ are similar. Prove that the plane $\Pi$ is parallel to the plane containing the base $A_1A_2\ldots A_n$.



Laurentiu Panaitopol
1997 Romania Team Selection Test
Suppose that $A$ be the set of all positive integer that can write in form $a^2+2b^2$ (where $a,b\in\mathbb {Z}$ and $b$ is not equal to $0$). Show that if $p$ be a prime number and $p^2\in A$ then $p\in A$.



Marcel Tena
1997 Romania Team Selection Test
Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation.
1997 Romania Team Selection Test
Let $p,q,r$ be distinct prime numbers and let

$$A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} $$

Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$, has elements $x$ and $y$ such that $x$ divides $y$.



Ioan Tomescu
1997 Romania Team Selection Test
Let $ABCDEF$ be a convex hexagon, and let $P= AB \cap CD$, $Q = CD \cap EF$, $R = EF \cap AB$, $S = BC \cap DE$, $T = DE \cap FA$, $U = FA \cap BC$. Prove that



$\frac{PQ}{CD} = \frac{QR}{EF} = \frac{RP}{AB}$ if and only if $\frac{ST}{DE} = \frac{TU}{FA} = \frac{US}{BC}$
1997 Romania Team Selection Test
Let $P$ be the set of points in the plane and $D$ the set of lines in the plane.  Determine whether there exists a bijective function $f: P \rightarrow D$ such that for any three collinear points $A$, $B$, $C$, the lines $f(A)$, $f(B)$, $f(C)$ are either parallel or concurrent.



Gefry Barad
1997 Romania Team Selection Test
Find all functions $f: \mathbb{R}\to [0;+\infty)$ such that:

$$f(x^2+y^2)=f(x^2-y^2)+f(2xy)$$for all real numbers $x$ and $y$.



Laurentiu Panaitopol
1997 Romania Team Selection Test
Let $n\ge 2$ be an integer and let $P(X)=X^n+a_{n-1}X^{n-1}+\ldots +a_1X+1$ be a polynomial with positive integer coefficients. Suppose that $a_k=a_{n-k}$ for all $k\in 1,2,\ldots,n-1$. Prove that there exist infinitely many pairs of positive integers $x,y$ such that $x|P(y)$ and $y|P(x)$.



Remus Nicoara
1997 Romania Team Selection Test
Let $P(X),Q(X)$ be monic irreducible polynomials with rational coefficients. suppose that $P(X)$ and $Q(X)$ have roots $\alpha$ and $\beta$ respectively, such that $\alpha + \beta $ is rational. Prove that $P(X)^2-Q(X)^2$ has a rational root.



Bogdan Enescu
1997 Romania Team Selection Test
Let $a>1$ be a positive integer. Show that the set of integers

$$\{a^2+a-1,a^3+a^2-1,\ldots ,a^{n+1}+a^n-1,\ldots\}$$

contains an infinite subset of pairwise coprime integers.



Mircea Becheanu
1997 Romania Team Selection Test
The vertices of a regular dodecagon are coloured either blue or red. Find the number of all possible colourings which do not contain monochromatic sub-polygons.



Vasile Pop
1997 Romania Team Selection Test
Let $w$ be a circle and $AB$ a line not intersecting $w$. Given a point $P_{0}$ on $w$, define the sequence $P_{0},P_{1},\ldots $ as follows: $P_{n+1}$ is the second intersection with $w$ of the line passing through $B$ and the second intersection of the line $AP_{n}$ with $w$. Prove that for a positive integer $k$, if $P_{0}=P_{k}$ for some choice of $P_{0}$, then $P_{0}=P_{k}$ for any choice of $P_{0}$.



Gheorge Eckstein
1996 Romania Team Selection Test
Let $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ be a function such that for every regular $ n $-gon $ A_1A_2 \ldots A_n $ we have $ f(A_1)+f(A_2)+\cdots +f(A_n)=0 $. Prove that $ f(x)=0 $ for all reals $ x $.
1996 Romania Team Selection Test
Find the greatest positive integer $n$ for which there exist $n$ nonnegative integers $x_1, x_2,\ldots , x_n$, not all zero, such that for any $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$ from the set $\{-1, 0, 1\}$, not all zero, $\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n$ is not divisible by $n^3$.
1996 Romania Team Selection Test
Let $ x,y\in \mathbb{R} $. Show that if  the set $ A_{x,y}=\{ \cos {(n\pi x)}+\cos {(n\pi y)} \mid n\in \mathbb{N}\} $ is finite then $ x,y \in \mathbb{Q} $.



Vasile Pop
1996 Romania Team Selection Test
Let $ ABCD $ be a cyclic quadrilateral and let $ M $ be the set of incenters and excenters of the triangles $ BCD $, $ CDA $, $ DAB $, $ ABC $ (so 16 points in total). Prove that there exist two sets $ \mathcal{K} $ and $ \mathcal{L} $ of four parallel lines each, such that every line in $ \mathcal{K} \cup \mathcal{L} $  contains exactly four points of $ M $.
1996 Romania Team Selection Test
Let $A$ and $B$ be points on a circle $\mathcal{C}$ with center $O$ such that $\angle AOB = \dfrac {\pi}2$. Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ at $A$ and $B$ respectively and are also externally tangent to one another. The circle $\mathcal{C}_3$ lies in the interior of $\angle AOB$ and it is tangent  externally to $\mathcal{C}_1$, $\mathcal{C}_2$ at $P$ and $R$ and internally tangent to $\mathcal{C}$ at $S$. Evaluate the value of  $\angle PSR$.
1996 Romania Team Selection Test
Let $ a\in \mathbb{R} $ and $ f_1(x),f_2(x),\ldots,f_n(x): \mathbb{R} \rightarrow \mathbb{R} $ are the additive functions such that for every $ x\in \mathbb{R} $ we have $ f_1(x)f_2(x) \cdots f_n(x) =ax^n $. Show that there exists $ b\in \mathbb {R} $ and $ i\in {\{1,2,\ldots,n}\} $ such that for every $ x\in \mathbb{R} $ we have $ f_i(x)=bx $.
1996 Romania Team Selection Test
Let $ p_1,p_2,\ldots,p_k $ be the distinct prime divisors of $ n $ and let $ a_n=\frac {1}{p_1}+\frac {1}{p_2}+\cdots+\frac {1}{p_k} $ for $ n\geq 2 $. Show that for every positive integer $ N\geq 2 $ the following inequality holds:  $ \sum_{k=2}^{N} a_2a_3 \cdots a_k <1 $



Laurentiu Panaitopol
1996 Romania Team Selection Test
Let $ n\geq 3 $ be an integer and let $ x_1,x_2,\ldots,x_{n-1} $ be nonnegative integers such that



\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n \\ x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}



Find the minimal value of $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $.
1996 Romania Team Selection Test
Let $ n $ and $ r $ be positive integers and $ A $ be a set of lattice points in the plane such that any open disc of radius $ r $ contains a point of $ A $. Show that

for any coloring of the points of $ A $ in $ n $ colors there exists four points of the same color which are the vertices of a rectangle.
1996 Romania Team Selection Test
Find all primes $ p,q $ such that $ \alpha^{3pq} -\alpha \equiv 0 \pmod {3pq} $ for all integers $ \alpha $.
1996 Romania Team Selection Test
Let $ n\geq 3 $ be an integer and let $ p\geq 2n-3 $ be a prime number. For a set $ M $ of $ n $ points in the plane, no 3 collinear, let $ f: M\to \{0,1,\ldots, p-1\} $ be a function such that



(i) exactly one point of $ M $ maps to 0,

(ii) if a circle $ \mathcal{C} $ passes through 3 distinct points of $ A,B,C\in M $ then $ \sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p $.



Prove that all the points in $ M $ lie on a circle.
1996 Romania Team Selection Test
Let $ x_1,x_2,\ldots,x_n $ be positive real numbers and $ x_{n+1} = x_1 + x_2 + \cdots + x_n $. Prove that

$$ \sum_{k=1}^n \sqrt { x_k (x_{n+1} - x_k)} \leq \sqrt { \sum_{k=1}^n x_{n+1}(x_{n+1}-x_k)}. $$

Mircea Becheanu
1996 Romania Team Selection Test
Let $ x,y,z $ be real numbers. Prove that the following conditions are equivalent:



(i) $ x,y,z $ are positive numbers and $ \dfrac 1x + \dfrac 1y + \dfrac 1z \leq 1 $;

(ii) $ a^2x+b^2y+c^2z>d^2 $ holds for every quadrilateral with sides $ a,b,c,d $.
1996 Romania Team Selection Test
Let $ S $ be a set of $ n $ concentric circles in the plane. Prove that if a function $ f: S\to S $ satisfies the property

$$ d( f(A),f(B)) \geq d(A,B) $$ for all $ A,B \in S $, then $ d(f(A),f(B)) = d(A,B) $, where $ d $ is the euclidean distance function.
1996 Romania Team Selection Test
Let $ n\geq 3 $ be an integer and let $ \mathcal{S} \subset \{1,2,\ldots, n^3\} $ be a set with $ 3n^2 $ elements. Prove that there exist nine distinct numbers  $ a_1,a_2,\ldots,a_9 \in \mathcal{S} $ such that the following system has a solution in nonzero integers:

\begin{eqnarray*} a_1x + a_2y +a_3 z &=& 0 \\ a_4x + a_5 y + a_6 z &=& 0 \\ a_7x + a_8y + a_9z &=& 0. \end{eqnarray*}



Marius Cavachi
1995 Romania Team Selection Test
Let AD  be the altitude of a triangle  ABC  and  E , F  be the incenters of the triangle  ABD  and  ACD , respectively. line  EF  meets  AB  and AC at  K  and L. prove tht AK=AL if and only if AB=AC or A=90
1995 Romania Team Selection Test
Find all positive integers $ x,y,z,t$ such that $ x,y,z$ are pairwise coprime and $ (x + y)(y + z)(z + x) = xyzt$.
1995 Romania Team Selection Test
Let $n \geq 6$ and $3 \leq p < n - p$ be two integers. The vertices of a regular $n$-gon are colored so that $p$ vertices are red and the others are black. Prove that there exist two congruent polygons with at least $[p/2] + 1$ vertices, one with all the vertices red and the other with all the vertices black.
1995 Romania Team Selection Test
Let $m,n$ be positive integers, greater than 2.Find the number of polynomials of degree $2n-1$ with distinct coefficients from the set $\left\{ 1,2,\ldots,m\right\}$ which are divisible by $x^{n-1}+x^{n-2}+\ldots+1.$
1995 Romania Team Selection Test
The sequence $ (x_n)$ is defined by $ x_1=1,x_2=a$ and $ x_n=(2n+1)x_{n-1}-(n^2-1)x_{n-2}$ $ \forall n \geq 3$, where $ a \in N^*$.For which value of $ a$ does the sequence have the property that $ x_i|x_j$ whenever $ i<j$.
1995 Romania Team Selection Test
Suppose that $n$ polygons of area $s = (n - 1)^2$ are placed on a polygon of area $S = \frac{n(n - 1)^2}{2}$. Prove that there exist two of the $n$ smaller polygons whose intersection has the area at least $1$.
1995 Romania Team Selection Test
Let $M, N, P, Q$ be points on sides $AB, BC, CD, DA$ of a convex quadrilateral $ABCD$ such that $AQ = DP = CN = BM$. Prove that if $MNPQ$ is a square, then $ABCD$ is also a square.
1995 Romania Team Selection Test
A convex set $S$ on a plane, not lying on a line, is painted in $p$ colors.

Prove that for every $n \ge 3$ there exist infinitely many congruent $n$-gons whose vertices are of the same color.
1995 Romania Team Selection Test
Let $a_1, a_2,...., a_n$ be distinct positive integers.

Prove that $(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) \ge  2(a_1^3 + ...+ a_n^3)^2$ and find the cases of equality.
1995 Romania Team Selection Test
A cube is partitioned into finitely many rectangular parallelepipeds with the edges parallel to the edges of the cube. Prove that if the sum of the volumes of the circumspheres of these parallelepipeds equals the volume of the circumscribed sphere of the cube, then all the parallelepipeds are cubes.
1995 Romania Team Selection Test
Let $f$ be an irreducible (in $Z[x]$) monic polynomial with integer coefficients and of odd degree greater than $1$. Suppose that the modules of the roots of $f$ are greater than $1$ and that $f(0)$ is a square-free number. Prove that

the polynomial $g(x) = f(x^3)$ is also irreducible
1995 Romania Team Selection Test
Find a sequence of positive integers $f(n)$ ($n \in \mathbb{N}$) such that:

(i) $f(n) \leq n^8$ for any $n \geq 2$;

(ii) for any distinct $a_1, \cdots, a_k, n$, $f(n) \neq f(a_1) + \cdots+ f(a_k)$.
1995 Romania Team Selection Test
How many colorings of an $n$-gon in $p \ge 2$ colors are there such that no two neighboring vertices have the same color?
1995 Romania Team Selection Test
For each positive integer $ n$,define $ f(n)=lcm(1,2,...,n)$.

(a)Prove that for every $ k$ there exist $ k$ consecutive positive integers on which $ f$ is constant.

(b)Find the maximum possible cardinality of a set of consecutive positive integers on which $ f$ is strictly increasing and find all sets for which this maximum is attained.
1995 Romania Team Selection Test
The altitudes of a triangle have integer length and its inradius is a prime number. Find all possible values of the sides of the triangle.
1995 Romania Team Selection Test
Let $ABCD$ be a convex quadrilateral. Suppose that similar isosceles triangles $APB, BQC, CRD, DSA$ with the bases on the sides of $ABCD$ are constructed in the exterior of the quadrilateral such that $PQRS$ is a rectangle but not a square. Show that $ABCD$ is a rhombus.
1994 BMO TST – Romania
Prove that if $n$ is a square-free positive integer, there are no coprime positive integers $x$ and $y$ such that $(x + y)^3$ divides $x^n+y^n$
1994 BMO TST – Romania
Let $n\geq  4$ be an integer. Find the maximum possible area of an $n-gon$ inscribed in a unit cicle and having two perpendicular diagonals.
1994 BMO TST – Romania
Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.
1994 BMO TST – Romania
Consider a tetrahedron$ A_1A_2A_3A_4$. A point $N$ is said to be a Servais point if its projections on the six edges of the tetrahedron lie in a plane $\alpha(N)$ (called Servais plane). Prove that if all the six points $Nij$ symmetric to a point $M$ with respect to the midpoints $Bij$ of the edges $A_iA_j$ are Servais points, then $M$ is contained in all Servais planes $\alpha(Nij )$
1994 Romania TST for IMO
Let $ X_n=\{1,2,...,n\}$,where $ n \geq 3$.

We define the measure $ m(X)$ of $ X\subset X_n$ as the sum of its elements.(If $ |X|=0$,then $ m(X)=0$).

A set $ X \subset X_n$ is said to be even(resp. odd) if $ m(X)$ is even(resp. odd).

(a)Show that the number of even sets equals the number of odd sets.

(b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets.

(c)Compute the sum of the measures of the odd sets.
1994 Romania TST for IMO
Let $ n$ be an odd positive integer. Prove that $((n-1)^n+1)^2$ divides $ n(n-1)^{(n-1)^n+1}+n$.
1994 Romania TST for IMO
Prove that the sequence $a_n = 3^n- 2^n$  contains no three numbers in geometric progression.
1994 Romania TST for IMO
Inscribe an equilateral triangle of minimum side in a given acute-angled triangle $ABC$ (one vertex on each side).
1994 Romania TST for IMO
First of all, I'm pretty sure $a_{n+1}$ is supposed to be $\frac{a_n}2$ when $a_n$ is even, not $n$ (and $a_n+7$ when $a_n$ is odd). Second of all, I think $a_1=1993^{1994^{1995}}$, that is, the topmost number is $1995$, not $1994$.



Every number $>7$ is turned into one $<7$ in at most $2$ steps, so the minimum is $\le 7$. Since our initial term is not divisible by $7$ and it's clear that the rules can't produce terms divisible by $7$ when there are none, it means that the minimum is $\le 6$. $6,5,3$ go to $3$, while $4,2,1$ go to $1$, so there are two possibilities: the minimum is either $3$ or $1$.



$2$ is a quadratic residue modulo $7$, so either all the terms of the sequence are quadratic residues modulo $7$, or none are. Since the initial term is $4\pmod 7$, it means that all terms are quadratic residues of $7$, so the minimum can't be $3$, meaning that it must be $1$.
1994 Romania TST for IMO
Let $S_1, S_2,S_3$ be spheres of radii $a, b, c$ respectively whose centers lie on a line $l$. Sphere $S_2$ is externally tangent to $S_1$ and $S_3$, whereas $S_1$ and $S_3$ have no common points. A straight line t touches each of the spheres,

Find the sine of the angle between $l$ and $t$
1994 Romania TST for IMO
Let $a_1, a_2, . . ., a_n$ be a finite sequence of $0$ and $1$. Under any two consecutive terms of this sequence $0$ is written if the digits are equal and $1$ is written otherwise. This way a new sequence of length $n -1$ is obtained.

By repeating this procedure $n - 1$ times one obtains a triangular table of $0$ and $1$. Find the maximum possible number of ones that can appear on this table
1994 Romania TST for IMO
Let be given two concentric circles of radii $R$ and $R_1 > R$. Let quadrilateral $ABCD$ is inscribed in the smaller circle and let the rays $CD, DA, AB, BC$ meet the larger circle at $A_1, B_1, C_1, D_1$ respectively.

Prove that

$$ \frac{\sigma(A_1B_1C_1D_1)}{\sigma(ABCD)} \geq \frac{R_1^2}{R^2}$$where $\sigma(P)$ denotes the area of a polygon $P.$
1994 Romania TST for IMO
Let $p$ be a (positive) prime number. Suppose that real numbers $a_1, a_2, . . ., a_{p+1}$ have the property that, whenever one of the numbers is deleted, the remaining numbers can be partitioned into two classes with the same arithmetic mean. Show that these numbers must be equal.
1994 Romania TST for IMO
Let $n$ be a positive integer. Find the number of polynomials $P(x)$ with coefficients in $\{0, 1, 2, 3\}$ for which $P(2) = n$.
1994 Romania TST for IMO
Determine all integer solutions of the equation $x^n+y^n=1994$ where $n\geq 2$
1994 Romania TST for IMO
Find a sequence of positive integer $f(n)$, $n \in \mathbb{N}$ such that



$(1)$ $f(n) \leq n^8$ for any $n \geq 2$,

$(2)$ for any pairwisely distinct natural numbers $a_1,a_2,\cdots, a_k$ and $n$, we have that

$$f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)$$
1993 Romania Team Selection Test
Consider the sequence $z_n = (1+i)(2+i)...(n+i)$.

Prove that the sequence $Im$ $z_n$ contains infinitely many positive and infinitely many negative numbers.
1993 Romania Team Selection Test
Let $ABC$ be a triangle inscribed in the circle $\mathcal{C}(O,R)$ and circumscribed to the circle $\mathcal{C}(L,r)$. Denote $d=\dfrac{Rr}{R+r}$. Show that there exists a triangle $DEF$ such that for any interior point $M$ in $ABC$ there exists a point $X$ on the sides of $DEF$ such that $MX\le d$.



Dan Brânzei
1993 Romania Team Selection Test
Show that the set $\{1,2,....,2^n\}$ can be partitioned in two classes, none of which contains an arithmetic progression of length $2n$.
1993 Romania Team Selection Test
Prove that the equation $ (x+y)^n=x^m+y^m$ has a unique solution in integers with $ x>y>0$ and $ m,n>1$.
1993 Romania Team Selection Test
Find max. numbers $A$ wich is true ineq.:



$\frac{x}{\sqrt{y^{2}+z^{2}}}+\frac{y}{\sqrt{x^{2}+z^{2}}}+\frac{z}{\sqrt{x^{2}+y^{2}}}\geq A$



$x,y,z$ are positve reals numberes! :wink:
1993 Romania Team Selection Test
$ x^2 + y^2 + z^2 = 1993$ then prove $ x + y + z$ can't be a perfect square:
1993 Romania Team Selection Test
Suppose that each of the diagonals $AD,BE,CF$ divides the hexagon $ABCDEF$ into two parts of the same area and perimeter. Does the hexagon necessarily have a center of symmetry?
1993 Romania Team Selection Test
Let $Y$ be the family of all subsets of $X = \{1,2,...,n\}$ ($n > 1$) and let $f : Y \to X$ be an arbitrary mapping. Prove that there exist distinct subsets $A,B$ of $X$ such that $f(A) = f(B) = max A\triangle B$, where $A\triangle B = (A-B)\cup(B-A)$.
1993 Romania Team Selection Test
Let $f : R^+ \to R$ be a strictly increasing function such that $f\left(\frac{x+y}{2}\right) < \frac{f(x)+ f(y)}{2}$ for all $x,y > 0$.

Prove that the sequence $a_n = f(n)$ ($n \in N$) does not contain an infinite arithmetic progression.
1993 Romania Team Selection Test
For coprime integers $m > n > 1$ consider the polynomials $f(x) = x^{m+n} -x^{m+1} -x+1$ and $g(x) = x^{m+n} +x^{n+1} -x+1$. If $f$ and $g$ have a common divisor of degree greater than $1$, find this divisor.
1993 Romania Team Selection Test
Find all integers $n > 1$ for which there is a set $B$ of $n$ points in the plane such that for any $A \in B$ there are three points $X,Y,Z \in B$ with $AX = AY = AZ = 1$.
1993 Romania Team Selection Test
For each integer $n > 3$ find all quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $n_1 +n_2 +n_3 +n_4 = n$ which maximize the expression $$\frac{n!}{n_1!n_2!n_3!n_4!}2^{ {n_1 \choose 2}+{n_2 \choose 2}+{n_3 \choose 2}+{n_4 \choose 2}+n_1n_2+n_2n_3+n_3n_4}$$
1993 Romania Team Selection Test
Define the sequence ($x_n$) as follows: the first term is $1$, the next two are $2,4$, the next three are $5,7,9$, the next four are $10,12,14,16$, and so on. Express $x_n$ as a function of $n$.
1993 Romania Team Selection Test
Suppose that $ D,E,F$ are points on sides $ BC,CA,AB$ of a triangle $ ABC$ respectively such that $ BD=CE=AF$ and $ \angle BAD=\angle CBE=\angle ACF$.Prove that the triangle $ ABC$ is equilateral.
1993 Romania Team Selection Test
Let $ p\geq 5$ be a prime number.Prove that for any partition of the set $ P=\{1,2,3,...,p-1\}$ in $ 3$ subsets there exists numbers $ x,y,z$ each belonging to a distinct subset,such that $ x+y\equiv z (mod p)$
1992 Romania Team Selection Test
Suppose that$ f : N \to N$ is an increasing function such that $f(f(n)) = 3n$ for all $n$. Find $f(1992)$.
1992 Romania Team Selection Test
For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix}

a^2+1 & a  \\
a & 1  

\end{pmatrix}^{n-2}$
1992 Romania Team Selection Test
Let $ABCD$ be a tetrahedron; $B', C', D'$ be the midpoints of the edges $AB, AC, AD$; $G_A, G_B, G_C, G_D$ be the barycentres of the triangles $BCD, ACD, ABD, ABC$, and $G$ be the barycentre of the tetrahedron. Show that $A, G, G_B, G_C, G_D$ are all on a sphere if and only if $A, G, B', C', D'$ are also on a sphere.



Dan Brânzei
1992 Romania Team Selection Test
Let $x_1,x_2,...,x_n$ be real numbers with $1 \ge  x_1 \ge x_2\ge ... \ge x_n \ge 0$ and $x_1^2 +x_2^2+...+x_n^2= 1$.

If $[x_1 +x_2 +...+x_n] = m$, prove that $x_1 +x_2 +...+x_m \ge 1$.
1992 Romania Team Selection Test
Let $S > 1$ be a real number. The Cartesian plane is partitioned into rectangles whose sides are parallel to the axes of the coordinate system. and whose vertices have integer coordinates. Prove that if the area of each triangle if at most $S$, then for any positive integer $k$ there exist $k$ vertices of these rectangles which lie on a line.