Navvye/Olympiad-Problems-CSV huggingface dataset:
1992 Romania Team Selection Test
Let $ a_1, a_2, ..., a_k $ be distinct positive integers such that the $2^k$ sums $\displaystyle\sum\limits_{i=1}^{k}{\epsilon_i a_i}$, $\epsilon_i\in\left\{0,1\right\}$ are distinct.

a) Show that $ \dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_k}\le2(1-2^{-k}) $;

b) Find the sequences $(a_1,a_2,...,a_k)$ for which the equality holds.



Șerban Buzețeanu
1992 Romania Team Selection Test
Let $\pi$ be the set of points in a plane and $f : \pi \to \pi$ be a mapping such that the image of any triangle (as its polygonal line) is a square. Show that $f(\pi)$ is a square.
1992 Romania Team Selection Test
Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows:  The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$
1992 Romania Team Selection Test
Let $O$ be the circumcenter of an acute triangle $ABC$. Suppose that the circumradius of the triangle is $R = 2p$, where $p$ is a prime number. The lines $AO,BO,CO$ meet the sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Given that the lengths of $OA_1,OB_1,OC_1$ are positive integers, find the side lengths of the triangle.
1992 Romania Team Selection Test
Let $m,n$ be positive integers and $p$ be a prime number.

Show that if $\frac{7^m + p \cdot 2^n}{7^m - p \cdot  2^n}$ is an integer, then it is a prime number.
1992 Romania Team Selection Test
Let $(a_{n})_{n\geq 1}$ and $(b_{n})_{n\geq 1}$ be the sequence of positive integers defined by  $a_{n+1}=na_{n}+1$ and $b_{n+1}=nb_{n}-1$ for $n\geq 1$. Show that the two sequence cannot have infinitely many common terms.



Laurentiu Panaitopol
1992 Romania Team Selection Test
Let $m,n \ge 2$ be integers. The sides $A_{00}A_{0m}$ and $A_{nm}A_{n0}$ of a convex quadrilateral $A_{00}A_{0m}A_{nm}A_{n0}$ are divided into $m$ equal segments by points $A_{0j}$ and $A_{nj}$ respectively ($j = 1,...,m-1$). The other two sides are divided into $n$ equal segments by points $A_{i0}$ and $A_{im}$ ($i = 1,...,n -1$). Denote by $A_{ij}$ the intersection of lines $A_{0j}A{nj}$ and $A_{i0}A_{im}$, by $S_{ij}$ the area of quadrilateral $A_{ij}A_{i, j+1}A_{i+1, j+1}A_{i+1, j}$ and by $S$ the area of the big quadrilateral. Show that $S_{ij} +S_{n-1-i,m-1-j} =

\frac{2S}{mn}$
1992 Romania Team Selection Test
Let $x, y$ be real numbers such that $1\le x^2-xy+y^2\le2$. Show that:

a) $\dfrac{2}{9}\le x^4+y^4\le 8$;

b) $x^{2n}+y^{2n}\ge\dfrac{2}{3^n}$, for all $n\ge3$.



Laurențiu Panaitopol and Ioan Tomescu
1992 Romania Team Selection Test
In a tetrahedron $VABC$, let $I$ be the incenter and $A',B',C'$ be arbitrary points on the edges $AV,BV,CV$, and let $S_a,S_b,S_c,S_v$ be the areas of triangles $VBC,VAC,VAB,ABC$, respectively. Show that points $A',B',C',I$ are coplanar if and only if $\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v$
1992 Romania Team Selection Test
In the Cartesian plane is given a polygon $P$ whose vertices have integer coordinates and with sides parallel to the coordinate axes. Show that if the length of each edge of $P$ is an odd integer, then the surface of P cannot be partitioned into $2\times 1$ rectangles.
1991 Romania Team Selection Test
Suppose that $ a,b$ are positive integers for which $ A=\frac{a+1}{b}+\frac{b}{a}$ is an integer.Prove that $ A=3$.
1991 Romania Team Selection Test
Let $A_1A_2A_3A_4$ be a tetrahedron. For any permutation $(i, j,k,h)$ of $1,2,3,4$ denote:

- $P_i$ – the orthogonal projection of $A_i$ on $A_jA_kA_h$;

- $B_{ij}$ – the midpoint of the edge $A_iAj$,

- $C_{ij}$ – the midpoint of segment $P_iP_j$

- $\beta_{ij}$– the plane $B_{ij}P_hP_k$

- $\delta_{ij}$ – the plane $B_{ij}P_iP_j$

- $\alpha_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_kA_h$

- $\gamma_{ij}$  – the plane through $C_{ij}$ orthogonal to $A_iA_j$.

Prove that if the points $P_i$ are not in a plane, then the following sets of planes are concurrent:

(a) $\alpha_{ij}$, (b) $\beta_{ij}$, (c) $\gamma_{ij}$, (d) $\delta_{ij}$.
1991 Romania Team Selection Test
Prove the following identity for every $ n\in N$:

$ \sum_{j+h=n,j\geq h}\frac{(-1)^h2^{j-h}\binom{j}{h}}{j}=\frac{2}{n}$
1991 Romania Team Selection Test
A sequence $(a_n)$ of positive integers satisfies$(a_m,a_n) = a_{(m,n)}$ for all $m,n$.

Prove that there is a unique sequence $(b_n)$ of positive integers such that $a_n = \prod_{d|n} b_d$
1991 Romania Team Selection Test
Let $M=\{A_{1},A_{2},\ldots,A_{5}\}$ be a set of five points in the plane such that the area of each triangle $A_{i}A_{j}A_{k}$, is greater than 3. Prove that there exists a triangle with vertices in $M$ and having the area greater than 4.



Laurentiu Panaitopol
1991 Romania Team Selection Test
The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer.

Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.
1991 Romania Team Selection Test
Let $C$ be a coloring of all edges and diagonals of a convex $n$−gon in red and blue  (in Romanian, rosu and albastru). Denote by $q_r(C)$ (resp. $q_a(C)$) the number of quadrilaterals having all its edges and diagonals red (resp. blue).

Prove: $ \underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}$
1991 Romania Team Selection Test
Let $S$ be the set of all polygonal areas in a plane. Prove that there is a function $f : S \to (0,1)$ which satisfies $f(S_1 \cup S_2) = f(S_1)+ f(S_2)$ for any $S1,S2 \in S$ which have common points only on their borders
1991 Romania Team Selection Test
In a triangle $A_1A_2A_3$, the excribed circles corresponding to sides $A_2A_3$, $A_3A_1$, $A_1A_2$ touch these sides at $T_1$, $T_2$, $T_3$, respectively. If $H_1$, $H_2$, $H_3$ are the orthocenters of triangles $A_1T_2T_3$, $A_2T_3T_1$, $A_3T_1T_2$, respectively, prove that lines $H_1T_1$, $H_2T_2$, $H_3T_3$ are concurrent.
1991 Romania Team Selection Test
Let $n \ge 3$ be an integer. A finite number of disjoint arcs with the total sum of length $1 -\frac{1}{n}$ are given on a circle of perimeter $1$. Prove that there is a regular $n$-gon whose all vertices lie on the considered arcs
1991 Romania Team Selection Test
Let $x_1,x_2,...,x_{2n}$ be positive real numbers with the sum $1$. Prove that

$$x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}$$
1991 Romania Team Selection Test
Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$.

If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.
1991 Romania Team Selection Test
The diagonals of a pentagon $ABCDE$ determine another pentagon $MNPQR$. If $MNPQR$ and $ABCDE$ are similar, must $ABCDE$ be regular?
1991 Romania Team Selection Test
Let $a_1<a_2<\cdots<a_n$ be positive integers. Some colouring of $\mathbb{Z}$ is periodic with period $t$ such that for each $x\in \mathbb{Z}$ exactly one of $x+a_1,x+a_2,\dots,x+a_n$ is coloured. Prove that $n\mid t$.



Andrei Radulescu-Banu
1990 Romania Team Selection Test
Let $f : N \to N$ be a function such that the set $\{k | f(k) < k\}$ is finite.

Prove that the set $\{k | g(f(k)) \le k\}$ is infinite for all functions $g : N \to N$.
1990 Romania Team Selection Test
Prove that in any triangle $ABC$ the following inequality holds: $$ \frac{a^{2}}{b+c-a}+\frac{b^{2}}{a+c-b}+\frac{c^{2}}{a+b-c}\geq 3\sqrt{3}R. $$



Laurentiu Panaitopol
1990 Romania Team Selection Test
Prove that for any positive integer $n$, the least common multiple of the numbers $1,2,\ldots,n$ and the least common multiple of the numbers: $$\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}$$ are equal if and only if $n+1$ is a prime number.



Laurentiu Panaitopol
1990 Romania Team Selection Test
Let $M$ be a point on the edge $CD$ of a tetrahedron $ABCD$ such that the tetrahedra $ABCM$ and $ABDM$ have the same total areas. We denote by $\pi_{AB}$ the plane $ABM$. Planes $\pi_{AC},...,\pi_{CD}$ are analogously defined. Prove that the six planes $\pi_{AB},...,\pi_{CD}$ are concurrent in a certain point $N$, and show that $N$ is symmetric to the incenter $I$ with respect to the barycenter $G$.
1990 Romania Team Selection Test
Let a,b,n be positive integers such that $(a,b) = 1$.

Prove that if $(x,y)$ is a solution of the equation $ax+by = a^n + b^n$ then

$$\left[\frac{x}{b}\right]+\left[\frac{y}{a}\right]=\left[\frac{a^{n-1}}{b}\right]+\left[\frac{b^{n-1}}{a}\right]$$
1990 Romania Team Selection Test
Prove the following equality for all positive integers $m,n$:

$$\sum_{k=0}^{n} {m+k \choose  k} 2^{n-k}  +\sum_{k=0}^m {n+k \choose k}2^{m-k}= 2^{m+n+1}$$
1990 Romania Team Selection Test
Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.
1990 Romania Team Selection Test
The six faces of a hexahedron are quadrilaterals. Prove that if seven its vertices lie on a sphere, then the eighth vertex also lies on the sphere.
1990 Romania Team Selection Test
Let $O$ be the circumcenter of an acute triangle $ABC$ and $R$ be its circumcenter. Consider the disks having $OA,OB,OC$ as diameters, and let $\Delta$ be the set of points in the plane belonging to at least two of the disks. Prove that the area of $\Delta$ is greater than $R^2/8$.
1990 Romania Team Selection Test
Prove that there are infinitely many n’s for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$, and prove that there are infinitely many $n$’s for which there is no such partition.
1990 Romania Team Selection Test
The sequence $ (x_n)_{n \geq 1}$ is defined by:

$ x_1=1$

$ x_{n+1}=\frac{x_n}{n}+\frac{n}{x_n}$

Prove that $ (x_n)$ increases and $ [x_n^2]=n$.
1990 Romania Team Selection Test
For a set $S$ of $n$ points, let $d_1 > d_2 >... > d_k > ...$ be the distances between the points.

A function $f_k: S \to N$ is called a coloring function if, for any pair $M,N$ of points in $S$ with $MN \le d_k$ , it takes the value $f_k(M)+ f_k(N)$ at some point. Prove that for each $m \in N$ there are positive integers $n,k$ and a set $S$ of $n$ points such that every coloring function $f_k$ of $S$ satisfies $| f_k(S)| \le m$
1990 Romania Team Selection Test
The distance between any two of six given points in the plane is at least $1$. Prove that the distance between some two points is at least  $\sqrt{\frac{5+\sqrt5}{2}}$
1990 Romania Team Selection Test
Let $p,q$ be positive prime numbers and suppose $q>5$. Prove that if $q \mid 2^{p}+3^{p}$, then $q>p$.



Laurentiu Panaitopol
1990 Romania Team Selection Test
In a group of $n$ persons,

(i) each person is acquainted to exactly $k$ others,

(ii) any two acquainted persons have exactly $l$ common acquaintances,

(iii) any two non-acquainted persons have exactly $m$ common acquaintances.

Prove that $m(n-k -1) = k(k -l -1)$.
1989 Romania Team Selection Test
Let $M$ denote the set of $m\times n$ matrices with entries in the set $\{0,1,2,3,4\}$ such that in each row and each column the sum of elements is divisible by $5$. Find the cardinality of set $M$.
1989 Romania Team Selection Test
Let $P$ be a point on a circle $C$ and let $\phi$ be a given angle incommensurable with $2\pi$. For each $n \in  N, P_n$ denotes the image of $P$ under the rotation about the center $O$ of $C$ by the angle $\alpha_n = n \phi$. Prove that the set $M = \{P_n | n \ge 0\}$ is dense in $C$.
1989 Romania Team Selection Test
Let $ABCD$ be a parallelogram and $M,N$ be points in the plane such that $C \in (AM)$ and $D  \in (BN)$. Lines $NA,NC$ meet lines $MB,MD$ at points $E,F,G,H$. Show that points $E,F,G,H$ lie on a circle if and only if $ABCD$ is a rhombus.
1989 Romania Team Selection Test
A family of finite sets $\left\{ A_{1},A_{2},.......,A_{m}\right\} $is called equipartitionable  if there is a function $\varphi:\cup_{i=1}^{m}$$\rightarrow\left\{ -1,1\right\} $ such that $\sum_{x\in A_{i}}\varphi\left(x\right)=0$ for every $i=1,.....,m.$ Let $f\left(n\right)$ denote the smallest possible number of $n$-element sets which form a non-equipartitionable family. Prove that

a) $f(4k +2) = 3$ for each nonnegative integer $k$,

b) $f\left(2n\right)\leq1+m d\left(n\right)$, where $m d\left(n\right)$ denotes the least positive non-divisor of $n.$
1989 Romania Team Selection Test
A laticial cycle of length $n$ is a sequence of lattice points $(x_k, y_k)$, $k = 0, 1,\cdots, n$, such that $(x_0, y_0) = (x_n, y_n) = (0, 0)$ and $|x_{k+1} -x_{k}|+|y_{k+1} - y_{k}| = 1$ for each $k$. Prove that for all $n$, the number of latticial cycles of length $n$ is a perfect square.
1989 Romania Team Selection Test
Let the sequence ($a_n$) be defined by $a_n = n^6 +5n^4 -12n^2 -36, n \ge 2$.

(a) Prove that any prime number divides some term in this sequence.

(b) Prove that there is a positive integer not dividing any term in the sequence.

(c) Determine the least $n \ge 2$ for which $1989 | a_n$.
1989 Romania Team Selection Test
Find all monic polynomials $P(x),Q(x)$ with integer coefficients such that $Q(0) =0$ and $P(Q(x)) = (x-1)(x-2)...(x-15)$.
1989 Romania Team Selection Test
Find all pair $(m,n)$ of integer ($m >1,n \geq 3$) with the following property:If an $n$-gon can be partitioned into $m$ isoceles triangles,then the $n$-gon has two congruent sides.
1989 Romania Team Selection Test
Let $r,n$ be positive integers. For a set $A$, let ${A \choose r}$ denote the family of all $r$-element subsets of $A$. Prove that if $A$ is infinite and $f : {A \choose r} \to {1,2,...,n}$ is any function, then there exists an infinite subset $B$ of $A$ such that $f(X) = f(Y)$ for all $X,Y \in {B \choose r}$.
1989 Romania Team Selection Test
Let $F$ be the set of all functions $f : N \to  N$ which satisfy $f(f(x))-2 f(x)+x = 0$ for all $x \in N$.

Determine the set $A =\{ f(1989) | f \in F\}$.
1989 Romania Team Selection Test
Let $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $a = nw- pv, b = pu-mw, c = mv-nu$.
1989 Romania Team Selection Test
(a) Find the point $M$ in the plane of triangle $ABC$ for which the sum $MA + MB+ MC$ is minimal.

(b) Given a parallelogram $ABCD$ whose angles do not exceed $120^o$, determine $min \{MA+ MB+NC+ND+ MN | M,N$ are in the plane $ABCD\}$ in terms of the sides and angles of the parallelogram.
1989 Romania Team Selection Test
Let $A,B,C$ be variable points on edges $OX,OY,OZ$ of a trihedral angle $OXYZ$, respectively.

Let $OA = a, OB = b, OC = c$ and $R$ be the radius of the circumsphere $S$ of $OABC$.

Prove that if points $A,B,C$ vary so that $a+b+c = R+l$, then the sphere $S$ remains tangent to a fixed sphere.
1989 Romania Team Selection Test
Prove that $\sqrt {1+\sqrt {2+\ldots +\sqrt {n}}}<2$, $\forall n\ge 1$.
1989 Romania Team Selection Test
The sequence ($a_n$) is defined by $a_1 = a_2 = 1, a_3 = 199$ and $a_{n+1} =\frac{1989+a_na_{n-1}}{a_{n-2}}$  for all $n \ge 3$. Prove that all terms of the sequence are positive integers
1989 Romania Team Selection Test
Let $F$ be the boundary and $M,N$ be any interior points of a triangle $ABC$. Consider the function $f_{M,N}: F \to R$ defined by $f_{M,N}(P) = MP^2 +NP^2$ and let $\eta_{M,N}$ be the number of points $P$ for which $f{M,N}$ attains its minimum.

(a) Prove that $1 \le \eta_{M,N} \le 3$.

(b) If $M$ is fixed, find the locus of $N$ for which $\eta_{M,N}  > 1$.

(c) Prove that the locus of $M$ for which there are points $N$ such that $\eta_{M,N}  = 3$ is the interior of a tangent hexagon.
1988 Romania Team Selection Test
Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone.



Octavian Stanasila
1988 Romania Team Selection Test
Let $OABC$ be a trihedral angle such that  $$ \angle BOC = \alpha, \quad \angle COA = \beta, \quad \angle AOB = \gamma , \quad \alpha + \beta + \gamma = \pi . $$ For any interior point $P$ of the trihedral angle let $P_1$, $P_2$ and $P_3$ be the projections of $P$ on the three faces. Prove that $OP \geq PP_1+PP_2+PP_3$.



Constantin Cocea
1988 Romania Team Selection Test
Consider all regular convex and star polygons inscribed in a given circle and having $n$ sides. We call two such polygons to be equivalent if it is possible to obtain one from the other using a rotation about the center of the circle. How many classes of such polygons exist?



Mircea Becheanu
1988 Romania Team Selection Test
Prove that for all positive integers $0<a_1<a_2<\cdots <a_n$ the following inequality holds: $$ (a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 . $$

Viorel Vajaitu
1988 Romania Team Selection Test
The cells of a $11\times 11$ chess-board are colored in 3 colors. Prove that there exists on the board a $m\times n$ rectangle such that the four cells interior to the rectangle and containing the four vertices of the rectangle have the same color.



Ioan Tomescu
1988 Romania Team Selection Test
Find all vectors of $n$ real numbers $(x_1,x_2,\ldots,x_n)$ such that

$$ \left\{ \begin{array}{ccc} x_1 & = & \dfrac 1{x_2} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n } \\ x_2 & = & \dfrac 1{x_1} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n} \\ \ & \cdots & \ \\ x_n & = & \dfrac 1{x_1} + \dfrac 1{x_2} + \cdots + \dfrac 1{x_{n-1}} \end{array} \right.  $$

Mircea Becheanu
1988 Romania Team Selection Test
In the plane there are given the lines $\ell_1$, $\ell_2$, the circle $\mathcal{C}$ with its center on the line $\ell_1$ and a second circle $\mathcal{C}_1$ which is tangent to $\ell_1$, $\ell_2$ and $\mathcal{C}$. Find the locus of the tangent point between $\mathcal{C}$ and $\mathcal{C}_1$ while the center of $\mathcal{C}$ is variable on $\ell_1$.



Mircea Becheanu
1988 Romania Team Selection Test
The positive integer $n$ is given and for all positive integers $k$, $1\leq k\leq n$, denote by $a_{kn}$ the number of all ordered sequences $(i_1,i_2,\ldots,i_k)$ of positive integers which verify the following two conditions:



a) $1\leq i_1<i_2< \cdots i_k \leq n$;



b) $i_{r+1}-i_r \equiv 1 \pmod 2$, for all $r \in\{1,2,\ldots,k-1\}$.



Compute the number $a(n) = \sum\limits_{k=1}^n a_{kn}$.



Ioan Tomescu
1988 Romania Team Selection Test
Prove that for all positive integers $n\geq 1$ the number $\prod^n_{k=1} k^{2k-n-1}$ is also an integer number.



Laurentiu Panaitopol.
1988 Romania Team Selection Test
Let $ p > 2$ be a prime number. Find the least positive number $ a$ which can be represented as

$$ a = (X - 1)f(X) + (X^{p - 1} + X^{p - 2} + \cdots + X + 1)g(X),

$$

where $ f(X)$ and $ g(X)$ are integer polynomials.



Mircea Becheanu.
1988 Romania Team Selection Test
Let $x,y,z$ be real numbers with $x+y+z=0$. Prove that $$ |\cos x |+ |\cos y| +| \cos z | \geq 1 .  $$ Viorel Vajaitu, Bogdan Enescu
1988 Romania Team Selection Test
The four vertices of a square are the centers of four circles such that the sum of theirs areas equals the square's area. Take an arbitrary point in the  interior of each circle. Prove that the four arbitrary points are the vertices of a convex quadrilateral.



Laurentiu Panaitopol
1988 Romania Team Selection Test
Let $a$ be a positive integer. The sequence $\{x_n\}_{n\geq 1}$ is defined by $x_1=1$, $x_2=a$ and $x_{n+2} = ax_{n+1} + x_n$ for all $n\geq 1$. Prove that $(y,x)$ is a solution of the equation $$ |y^2 - axy - x^2 | = 1 $$ if and only if there exists a rank $k$ such that $(y,x)=(x_{k+1},x_k)$.



Serban Buzeteanu
1988 Romania Team Selection Test
Let $\Delta$ denote the set of all triangles in a plane. Consider the function $f: \Delta\to(0,\infty)$ defined by $f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)$, for any triangle $ABC$ with $BC=a\leq CA=b\leq AB = c$. Find the set of values of $f$.
1988 Romania Team Selection Test
Let $[a,b]$ be a given interval of real numbers not containing integers. Prove that there exists $N>0$ such that $[Na,Nb]$ does not contain integer numbers and the length of the interval $[Na,Nb]$ exceedes $\dfrac 16$.
1988 Romania Team Selection Test
The finite sets $A_1$, $A_2$, $\ldots$, $A_n$ are given and we denote by $d(n)$ the number of elements which appear exactly in an odd number of sets chosen from $A_1$, $A_2$, $\ldots$, $A_n$. Prove that for any $k$, $1\leq k\leq n$ the number $${ d(n) - \sum\limits^n_{i=1} |A_i| + 2\sum\limits_{ i<j} |A_i \cap A_j | - \cdots + (-1)^k2^{k-1} \sum\limits_{i_1 <i_2 <\cdots < i_k} | A_{i_1} \cap A_{i_2} \cap \cdots \cap A_{i_k}}|  $$ is divisible by $2^k$.



Ioan Tomescu, Dragos Popescu
1987 Romania Team Selection Test
Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value.



Mircea Becheanu
1987 Romania Team Selection Test
Find all positive integers $A$ which can be represented in the form: $$ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) $$

where $m\geq n\geq p \geq 1$ are integer numbers.



Ioan Bogdan
1987 Romania Team Selection Test
Let $A$ be the set $A = \{ 1,2, \ldots, n\}$. Determine the maximum number of elements of a subset $B\subset A$ such that for all elements $x,y$ from $B$, $x+y$ cannot be divisible by $x-y$.



Mircea Lascu, Dorel Mihet
1987 Romania Team Selection Test
Let $ P(X) = a_{n}X^{n} + a_{n - 1}X^{n - 1} + \ldots + a_{1}X + a_{0}$ be a real polynomial of degree $ n$. Suppose $ n$ is an even number and:



a) $ a_{0} > 0$, $ a_{n} > 0$;



b) $ a_{1}^{2} + a_{2}^{2} + \ldots + a_{n - 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n - 1}$.



Prove that $ P(x)\geq 0$ for all real values $ x$.



Laurentiu Panaitopol
1987 Romania Team Selection Test
Let $A$ be the set $\{1,2,\ldots,n\}$, $n\geq 2$. Find the least number $n$ for which there exist permutations $\alpha$, $\beta$, $\gamma$, $\delta$ of the set $A$ with the property: $$ \sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) . $$

Marcel Chirita
1987 Romania Team Selection Test
The plane is covered with network of regular congruent disjoint hexagons. Prove that there cannot exist a square which has its four vertices in the vertices of the hexagons.



Gabriel Nagy
1987 Romania Team Selection Test
Determine all positive integers $n$ such that $n$ divides $3^n - 2^n$.
1987 Romania Team Selection Test
Let $ABCD$ be a square and $a$ be the length of his edges. The segments $AE$ and $CF$ are perpendicular on the square's plane in the same half-space and they have the length $AE=a$, $CF=b$ where $a<b<a\sqrt 3$. If $K$ denoted the set of the interior points of the square $ABCD$ determine $\min_{M\in K} \left( \max ( EM, FM ) \right) $ and $\max_{M\in K} \left( \min (EM,FM) \right)$.



Octavian Stanasila
1987 Romania Team Selection Test
Prove that for all real numbers $\alpha_1,\alpha_2,\ldots,\alpha_n$ we have $$ \sum_{i=1}^n \sum_{j=1}^n ij \cos {(\alpha_i - \alpha_j )} \geq 0. $$

Octavian Stanasila
1987 Romania Team Selection Test
Let $a,b,c$ be integer numbers such that $(a+b+c) \mid (a^{2}+b^{2}+c^{2})$. Show that there exist infinitely many positive integers $n$ such that $(a+b+c) \mid (a^{n}+b^{n}+c^{n})$.



Laurentiu Panaitopol
1987 Romania Team Selection Test
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations:  $$ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . $$ We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$, $x_1=\lambda y_1$, $\ldots$, $x_n= \lambda y_n$. How many nonequivalent solutions does the system have?



Mircea Becheanu