Navvye/Olympiad-Problems-CSV huggingface dataset:
2003 Romania Team Selection Test
Given is a rhombus $ABCD$ of side 1. On the sides $BC$ and $CD$ we are given the points $M$ and $N$ respectively, such that $MC+CN+MN=2$ and $2\angle MAN = \angle BAD$. Find the measures of the angles of the rhombus.



Cristinel Mortici
2003 Romania Team Selection Test
In a plane we choose a cartesian system of coordinates. A point $A(x,y)$ in the plane is called an integer point if and only if both $x$ and $y$ are integers. An integer point $A$ is called invisible if on the segment $(OA)$ there is at least one integer point.

Prove that for each positive integer $n$ there exists a square of side $n$ in which all the interior integer points are invisible.
2003 Romania Team Selection Test
Let $ABCDEF$ be a convex hexagon and denote by $A',B',C',D',E',F'$ the middle points of the sides $AB$, $BC$, $CD$, $DE$, $EF$ and $FA$ respectively. Given are the areas of the triangles $ABC'$, $BCD'$, $CDE'$, $DEF'$, $EFA'$ and $FAB'$. Find the area of the hexagon.



Kvant Magazine
2003 Romania Team Selection Test
A permutation $\sigma: \{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ is called straight if and only if for each integer $k$, $1\leq k\leq n-1$ the following inequality is fulfilled

$$ |\sigma(k)-\sigma(k+1)|\leq 2. $$

Find the smallest positive integer $n$ for which there exist at least 2003 straight permutations.



Valentin Vornicu
2003 Romania Team Selection Test
For every positive integer $n$ we denote by $d(n)$ the sum of its digits in the decimal representation. Prove that for each positive integer $k$ there exists a positive integer $m$ such that the equation $x+d(x)=m$ has exactly $k$ solutions in the set of positive integers.
2002 Romania Team Selection Test
Find all sets $A$ and $B$ that satisfy the following conditions:



a) $A \cup B= \mathbb{Z}$;



b) if $x \in A$ then $x-1 \in B$;



c) if $x,y \in B$ then $x+y \in A$.



Laurentiu Panaitopol
2002 Romania Team Selection Test
The sequence $ (a_n)$ is defined by: $ a_0=a_1=1$ and $ a_{n+1}=14a_n-a_{n-1}$ for all $ n\ge 1$.

Prove that $ 2a_n-1$ is a perfect square for any $ n\ge 0$.
2002 Romania Team Selection Test
Let $M$ and $N$ be the midpoints of the respective sides $AB$ and $AC$ of an acute-angled triangle $ABC$.  Let $P$ be the foot of the perpendicular from $N$ onto $BC$ and let $A_1$ be the midpoint of $MP$.  Points $B_1$ and $C_1$ are obtained similarly.  If $AA_1$, $BB_1$ and $CC_1$ are concurrent, show that the triangle $ABC$ is isosceles.



Mircea Becheanu
2002 Romania Team Selection Test
For any positive integer $n$, let $f(n)$ be the number of possible choices of signs $+\ \text{or}\ - $ in the algebraic expression $\pm 1\pm 2\ldots \pm n$, such that the obtained sum is zero. Show that $f(n)$ satisfies the following conditions:

a) $f(n)=0$ for $n=1\pmod{4}$ or $n=2\pmod{4}$.

b) $2^{\frac{n}{2}-1}\le f(n)\le 2^n-2^{\lfloor\frac{n}{2}\rfloor+1}$, for $n=0\pmod{4}$ or $n=3\pmod{4}$.



Ioan Tomsecu
2002 Romania Team Selection Test
Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$.



Dinu Șerbănescu
2002 Romania Team Selection Test
Let $P(x)$ and $Q(x)$ be integer polynomials of degree $p$ and $q$ respectively. Assume that $P(x)$ divides $Q(x)$ and all their coefficients are either $1$ or $2002$. Show that $p+1$ is a divisor of $q+1$.



Mihai Cipu
2002 Romania Team Selection Test
Let $a,b$ be positive real numbers. For any positive integer $n$, denote by $x_n$ the sum of digits of the number $[an+b]$ in it's decimal representation. Show that the sequence $(x_n)_{n\ge 1}$ contains a constant subsequence.



Laurentiu Panaitopol
2002 Romania Team Selection Test
At an international conference there are four official languages. Any two participants can speak in one of these languages. Show that at least $60\%$ of the participants can speak the same language.



Mihai Baluna
2002 Romania Team Selection Test
Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$.



Dinu Șerbănescu
2002 Romania Team Selection Test
Let $n\geq 4$ be an integer, and let $a_1,a_2,\ldots,a_n$ be positive real numbers such that $$ a_1^2+a_2^2+\cdots +a_n^2=1 . $$ Prove that the following inequality takes place $$ \frac{a_1}{a_2^2+1}+\cdots +\frac{a_n}{a_1^2+1} \geq \frac{4}{5}\left( a_1 \sqrt{a_1}+\cdots +a_n \sqrt{a_n} \right)^2 . $$ Bogdan Enescu, Mircea Becheanu
2002 Romania Team Selection Test
Let $n$ be a positive integer. $S$ is the set of nonnegative integers $a$ such that $1<a<n$ and $a^{a-1}-1$ is divisible by $n$. Prove that if $S=\{ n-1 \}$ then $n=2p$ where $p$ is a prime number.



Mihai Cipu and Nicolae Ciprian Bonciocat
2002 Romania Team Selection Test
Let $f:\mathbb{Z}\rightarrow\{ 1,2,\ldots ,n\}$ be a function such that $f(x)\not= f(y)$, for all $x,y\in\mathbb{Z}$ such that $|x-y|\in\{2,3,5\}$. Prove that $n\ge 4$.



Ioan Tomescu
2002 Romania Team Selection Test
Let $(a_n)_{n\ge 1}$ be a sequence of positive integers defined as $a_1,a_2>0$ and $a_{n+1}$ is the least prime divisor of $a_{n-1}+a_{n}$, for all $n\ge 2$.



Prove that a real number $x$ whose decimals are digits of the numbers $a_1,a_2,\ldots a_n,\ldots $ written in order, is a rational number.



Laurentiu Panaitopol
2002 Romania Team Selection Test
Find the least positive real number $r$ with the following property:



Whatever four disks are considered, each with centre on the edges of a unit square and the sum of their radii equals $r$, there exists an equilateral triangle which has its edges in three of the disks.



Radu Gologan
2002 Romania Team Selection Test
After elections, every parliament member (PM), has his own absolute rating. When the parliament set up, he enters in a group and gets a relative rating. The relative rating is the ratio of its own absolute rating to the sum of all absolute ratings of the PMs in the group. A PM can move from one group to another only if in his new group his relative rating is greater. In a given day, only one PM can change the group. Show that only a finite number of group moves is possible.



(A rating is positive real number.)
2002 Romania Team Selection Test
Let $m,n$ be positive integers of distinct parities and such that $m<n<5m$. Show that there exists a partition with two element subsets of the set $\{ 1,2,3,\ldots ,4mn\}$ such that the sum of numbers in each set is a perfect square.



Dinu Șerbănescu
2002 Romania Team Selection Test
Let $ABC$ be a triangle such that $AC\not= BC,AB<AC$ and let $K$ be it's circumcircle. The tangent to $K$ at the point $A$ intersects the line $BC$ at the point $D$. Let $K_1$ be the circle tangent to $K$ and to the segments $(AD),(BD)$. We denote by $M$ the point where $K_1$ touches $(BD)$. Show that $AC=MC$ if and only if $AM$ is the bisector of the $\angle DAB$.



Neculai Roman
2002 Romania Team Selection Test
There are $n$ players, $n\ge 2$, which are playing a card game with $np$ cards in $p$ rounds. The cards are coloured in $n$ colours and each colour is labelled with the numbers $1,2,\ldots ,p$. The game submits to the following rules:



each player receives

$p$ cards.

the player who begins the first round throws a card and each player has to discard a card of the same colour, if he has one; otherwise they can give an arbitrary card.

the winner of the round is the player who has put the greatest card of the same colour as the first one.

the winner of the round starts the next round with a card that he selects and the play continues with the same rules.

the played cards are out of the game.

Show that if all cards labelled with number $1$ are winners, then $p\ge 2n$.



Barbu Berceanu
2001 Romania Team Selection Test
Show that if $a,b,c$ are complex numbers that such that

$$ (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a$$then $a,b,c$ are real numbers.
2001 Romania Team Selection Test
a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function.



b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.
2001 Romania Team Selection Test
The sides of a triangle have lengths $a,b,c$. Prove that:

\begin{align*}(-a+b+c)(a-b+c)\, +\, & (a-b+c)(a+b-c)+(a+b-c)(-a+b+c)\\ &\le\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})\end{align*}
2001 Romania Team Selection Test
Three schools have $200$ students each. Every student has at least one friend in each school (if the student $a$ is a friend of the student $b$ then $b$ is a friend of $a$).

It is known that there exists a set $E$ of $300$ students (among the $600$) such that for any school $S$ and any two students $x,y\in E$ but not in $S$, the number of friends in $S$ of $x$ and $y$ are different.

Show that one can find a student in each school such that they are friends with each other.
2001 Romania Team Selection Test
Find all polynomials with real coefficients $P$ such that

$$ P(x)P(2x^2-1)=P(x^2)P(2x-1)$$

for every $x\in\mathbb{R}$.
2001 Romania Team Selection Test
The vertices $A,B,C$ and $D$ of a square lie outside a circle centred at $M$. Let $AA',BB',CC',DD'$ be tangents to the circle. Assume that the segments $AA',BB',CC',DD'$ are the consecutive sides of a quadrilateral $p$ in which a circle is inscribed. Prove that $p$ has an axis of symmetry.
2001 Romania Team Selection Test
Find the least $n\in N$ such that among any $n$ rays in space sharing a common origin there exist two which form an acute angle.
2001 Romania Team Selection Test
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.
2001 Romania Team Selection Test
Let $n$ be a positive integer and $f(x)=a_mx^m+\ldots + a_1X+a_0$, with $m\ge 2$, a polynomial with integer coefficients such that:

a) $a_2,a_3\ldots a_m$ are divisible by all prime factors of $n$,

b) $a_1$ and $n$ are relatively prime.

Prove that for any positive integer $k$, there exists a positive integer $c$, such that $f(c)$ is divisible by $n^k$.
2001 Romania Team Selection Test
Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called ideal if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n + p$ and $ n + q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$
2001 Romania Team Selection Test
Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.
2001 Romania Team Selection Test
Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that

$$f(x+y)\ge f(x)+yf(f(x)) $$

for every $x,y\in (0,\infty )$.
2001 Romania Team Selection Test
The tangents at $A$ and $B$ to the circumcircle of the acute triangle $ABC$ intersect the tangent at $C$ at the points $D$ and $E$, respectively. The line $AE$ intersects $BC$ at $P$ and the line $BD$ intersects $AC$ at $R$. Let $Q$ and $S$ be the midpoints of the segments $AP$ and $BR$ respectively. Prove that $\angle ABQ=\angle BAS$.
2001 Romania Team Selection Test
Consider a convex polyhedron $P$ with vertices $V_1,\ldots ,V_p$. The distinct vertices $V_i$ and $V_j$ are called neighbours if they belong to the same face of the polyhedron. To each vertex $V_k$ we assign a number $v_k(0)$, and  construct inductively the sequence $v_k(n)\ (n\ge 0)$ as follows: $v_k(n+1)$ is the average of the $v_j(n)$ for all neighbours $V_j$ of $V_k$ . If all numbers $v_k(n)$ are integers, prove that there exists the positive integer $N$ such that all $v_k(n)$ are equal for $n\ge N$ .
2000 Romania Team Selection Test
Let $n\ge 2$ be a positive integer. Find the number of functions $f:\{1,2,\ldots ,n\}\rightarrow\{1,2,3,4,5 \}$ which have the following property: $|f(k+1)-f(k)|\ge 3$, for any $k=1,2,\ldots n-1$.



Vasile Pop
2000 Romania Team Selection Test
Let $n\ge 1$ be a positive integer and $x_1,x_2\ldots ,x_n$ be real numbers such that $|x_{k+1}-x_k|\le 1$ for $k=1,2,\ldots ,n-1$. Prove that

$$\sum_{k=1}^n|x_k|-\left|\sum_{k=1}^nx_k\right|\le\frac{n^2-1}{4}$$



Gh. Eckstein
2000 Romania Team Selection Test
Prove that for any positive integers $n$ and $k$ there exist positive integers $a>b>c>d>e>k$ such that

$$n=\binom{a}{3}\pm\binom{b}{3}\pm\binom{c}{3}\pm\binom{d}{3}\pm\binom{e}{3}$$



Radu Ignat
2000 Romania Team Selection Test
Let $P_1P_2\ldots P_n$ be a convex polygon in the plane. We assume that for any arbitrary choice of vertices $P_i,P_j$ there exists a vertex in the polygon $P_k$ distinct from $P_i,P_j$ such that $\angle P_iP_kP_j=60^{\circ}$. Show that $n=3$.



Radu Todor
2000 Romania Team Selection Test
Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$.



Mihai Pitticari & Sorin Rǎdulescu
2000 Romania Team Selection Test
Determine all pairs $(m,n)$ of positive integers such that a $m\times n$ rectangle can be tiled with L-trominoes.
2000 Romania Team Selection Test
Let $a>1$ be an odd positive integer. Find the least positive integer $n$ such that $2^{2000}$ is a divisor of $a^n-1$.



Mircea Becheanu
2000 Romania Team Selection Test
Let $ABC$ be an acute-angled triangle and $M$ be the midpoint of the side $BC$. Let $N$ be a point in the interior of the triangle $ABC$ such that $\angle NBA=\angle BAM$ and $\angle NCA=\angle CAM$. Prove that $\angle NAB=\angle MAC$.



Gabriel Nagy
2000 Romania Team Selection Test
Let $S$ be the set of interior points of a sphere and $C$ be the set of interior points of a circle. Find, with proof, whether there exists a function $f:S\rightarrow C$ such that $d(A,B)\le d(f(A),f(B))$ for any two points $A,B\in S$ where $d(X,Y)$ denotes the distance between the points $X$ and $Y$.



Marius Cavachi
2000 Romania Team Selection Test
Let $P_1$ be a regular $n$-gon, where $n\in\mathbb{N}$. We construct $P_2$ as the regular $n$-gon whose vertices are the midpoints of the edges of $P_1$. Continuing analogously, we obtain regular $n$-gons $P_3,P_4,\ldots ,P_m$. For $m\ge n^2-n+1$, find the maximum number $k$ such that for any colouring of vertices of $P_1,\ldots ,P_m$ in $k$ colours there exists an isosceles trapezium $ABCD$ whose vertices $A,B,C,D$ have the same colour.



Radu Ignat
2000 Romania Team Selection Test
Let $P,Q$ be two monic polynomials with complex coefficients such that $P(P(x))=Q(Q(x))$ for all $x$. Prove that $P=Q$.



Marius Cavachi
2000 Romania Team Selection Test
Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.
1999 Romania Team Selection Test
a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11;



b) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11.
1999 Romania Team Selection Test
Let $ABC$ be an acute triangle. The interior angle bisectors of $\angle ABC$ and $\angle ACB$ meet the opposite sides in $L$ and $M$ respectively. Prove that there is a point $K$ in the interior of the side $BC$ such that the triangle $KLM$ is equilateral if and only if $\angle BAC = 60^\circ$.
1999 Romania Team Selection Test
Prove that for any positive integer $n$, the number

$$ S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n $$ is the sum of two consecutive perfect squares.



Dorin Andrica
1999 Romania Team Selection Test
Show that for all positive real numbers $x_1,x_2,\ldots,x_n$ with product 1, the following inequality holds

$$ \frac 1{n-1+x_1 } +\frac 1{n-1+x_2} + \cdots + \frac 1{n-1+x_n} \leq 1. $$
1999 Romania Team Selection Test
Let $x_1,x_2,\ldots,x_n$ be distinct positive integers. Prove that

$$ x_1^2+x_2^2 + \cdots + x_n^2 \geq \frac {2n+1}3 ( x_1+x_2+\cdots + x_n). $$

Laurentiu Panaitopol
1999 Romania Team Selection Test
Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.
1999 Romania Team Selection Test
Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that

$$ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . $$

Give an example of two such progressions having at least five terms.



Mihai Baluna
1999 Romania Team Selection Test
Let $a$ be a positive real number and $\{x_n\}_{n\geq 1}$ a sequence of real numbers such that $x_1=a$ and

$$ x_{n+1} \geq (n+2)x_n - \sum^{n-1}_{k=1}kx_k, \ \forall \ n\geq 1. $$

Prove that there exists a positive integer $n$ such that $x_n > 1999!$.



Ciprian Manolescu
1999 Romania Team Selection Test
Let $O,A,B,C$ be variable points in the plane such that $OA=4$, $OB=2\sqrt3$ and $OC=\sqrt {22}$. Find the maximum value of the area $ABC$.



Mihai Baluna
1999 Romania Team Selection Test
Let $a,n$ be integer numbers, $p$ a prime number such that $p>|a|+1$. Prove that the polynomial $f(x)=x^n+ax+p$ cannot be represented as a product of two integer polynomials.



Laurentiu Panaitopol
1999 Romania Team Selection Test
Let $n\geq 3$ and $A_1,A_2,\ldots,A_n$ be points on a circle. Find the largest number of acute triangles that can be considered with vertices in these points.



G. Eckstein
1999 Romania Team Selection Test
The participants to an international conference are native and foreign scientist. Each native scientist sends a message to a foreign scientist and each foreign scientist sends a message to a native scientist. There are native scientists who did not receive a message.



Prove that there exists a set $S$ of native scientists such that the outer $S$ scientists are exactly those who received messages from those foreign scientists who did not receive messages from scientists belonging to $S$.



Radu Niculescu
1999 Romania Team Selection Test
Let $X$ be a set with $n$ elements, and let $A_{1}$, $A_{2}$, ..., $A_{m}$ be subsets of $X$ such that:



1) $|A_{i}|=3$ for every $i\in\left\{1,2,...,m\right\}$;

2) $|A_{i}\cap A_{j}|\leq 1$ for all $i,j\in\left\{1,2,...,m\right\}$ such that $i \neq j$.



Prove that there exists a subset $A$ of $X$ such that $A$ has at least $\left[\sqrt{2n}\right]$ elements, and for every $i\in\left\{1,2,...,m\right\}$, the set $A$ does not contain $A_{i}$.



Alternative formulation. Let $X$ be a finite set with $n$ elements and $A_{1},A_{2},\ldots, A_{m}$ be three-elements subsets of $X$, such that $|A_{i}\cap A_{j}|\leq 1$, for every $i\neq j$. Prove that there exists $A\subseteq X$ with $|A|\geq \lfloor \sqrt{2n}\rfloor$, such that none of $A_{i}$'s is a subset of $A$.
1999 Romania Team Selection Test
A polyhedron $P$ is given in space. Find whether there exist three edges in $P$ which can be the sides of a triangle. Justify your answer!



Barbu Berceanu
1998 Romania Team Selection Test
A word of length $n$ is an ordered sequence $x_1x_2\ldots x_n$ where $x_i$ is a letter from the set $\{ a,b,c \}$. Denote by $A_n$ the set of words of length $n$ which do not contain any block $x_ix_{i+1}, i=1,2,\ldots ,n-1,$ of the form $aa$ or $bb$ and by $B_n$ the set of words of length $n$ in which none of the subsequences $x_ix_{i+1}x_{i+2}, i=1,2,\ldots n-2,$ contains all the letters $a,b,c$.

Prove that $|B_{n+1}|=3|A_n|$.



Vasile Pop
1998 Romania Team Selection Test
A parallelepiped has surface area 216 and volume 216. Show that it is a cube.
1998 Romania Team Selection Test
Let $m\ge 2$ be an integer. Find the smallest positive integer $n>m$ such that for any partition with two classes of the set $\{ m,m+1,\ldots ,n \}$ at least one of these classes contains three numbers $a,b,c$ (not necessarily different) such that $a^b=c$.



Ciprian Manolescu
1998 Romania Team Selection Test
Consider in the plane a finite set of segments such that the sum of their lengths is less than $\sqrt{2}$. Prove that there exists an infinite unit square grid covering the plane such that the lines defining the grid do not intersect any of the segments.



Vasile Pop
1998 Romania Team Selection Test
We are given an isosceles triangle $ABC$ such that $BC=a$ and $AB=BC=b$. The variable points $M\in (AC)$ and $N\in (AB)$ satisfy $a^2\cdot AM \cdot AN = b^2 \cdot BN \cdot CM$. The straight lines $BM$ and $CN$ intersect in $P$. Find the locus of the variable point $P$.



Dan Branzei
1998 Romania Team Selection Test
All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$.



Bogdan Enescu
1998 Romania Team Selection Test
Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.
1998 Romania Team Selection Test
Let $n\ge 2$ be an integer. Show that there exists a subset $A\in \{1,2,\ldots ,n\}$ such that:



i) The number of elements of $A$ is at most $2\lfloor\sqrt{n}\rfloor+1$



ii) $\{ |x-y| \mid x,y\in A, x\not= y\} = \{ 1,2,\ldots n-1 \}$



Radu Todor
1998 Romania Team Selection Test
An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.
1998 Romania Team Selection Test
Show that for any positive integer $n$ the polynomial $f(x)=(x^2+x)^{2^n}+1$ cannot be decomposed into the product of two integer non-constant polynomials.



Marius Cavachi
1998 Romania Team Selection Test
Let $ABC$ be an equilateral triangle and $n\ge 2$ be an integer. Denote by $\mathcal{A}$ the set of $n-1$ straight lines which are parallel to $BC$ and divide the surface $[ABC]$ into $n$ polygons having the same area and denote by $\mathcal{P}$ the set of $n-1$ straight lines parallel to $BC$ which divide the surface $[ABC]$ into $n$ polygons having the same perimeter.

Prove that the intersection $\mathcal{A} \cap \mathcal{P}$ is empty.



Laurentiu Panaitopol
1998 Romania Team Selection Test
Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that

$$ a_{1} + A_{1} = a_{2} + A_{2} = \cdots = a_{n} + A_{n},$$

where $ x + A$ denotes the set $ \{x + a \vert a \in A \}$.
1998 Romania Team Selection Test
Let $n$ be a positive integer and $\mathcal{P}_n$ be the set of integer polynomials of the form $a_0+a_1x+\ldots +a_nx^n$ where $|a_i|\le 2$ for $i=0,1,\ldots ,n$. Find, for each positive integer $k$, the number of elements of the set $A_n(k)=\{f(k)|f\in \mathcal{P}_n \}$.



Marian Andronache
1998 Romania Team Selection Test
Find all monotonic functions $u:\mathbb{R}\rightarrow\mathbb{R}$ which have the property that there exists a strictly monotonic function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that

$$f(x+y)=f(x)u(x)+f(y) $$

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



Vasile Pop
1998 Romania Team Selection Test
Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k + 1\}$, then  $ F(0) = F(1) = \ldots = F(k + 1)$.