Navvye/Olympiad-Problems-CSV huggingface dataset:
2001 Vietnam Team Selection Test
Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition



$$0 < a_{n+1} - a_n \leq 2001$$



for all $n \in \mathbb{N}^{*}$



Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.
2000 Vietnam Team Selection Test
Two circles $C_{1}$ and $C_{2}$ intersect at points $P$ and $Q$. Their common tangent, closer to $P$ than to $Q$, touches $C_{1}$ at $A$ and $C_{2}$ at $B$. The tangents to $C_{1}$ and $C_{2}$ at $P$ meet the other circle at points $E \not = P$ and $F \not = P$ , respectively. Let $H$ and $K$ be the points on the rays $AF$ and $BE$ respectively such that $AH = AP$ and $BK = BP$ . Prove that $A,H,Q,K,B$ lie on a circle.
2000 Vietnam Team Selection Test
Let $k$ be a given positive integer. Deﬁne $x_{1}= 1$ and, for each $n > 1$, set $x_{n+1}$ to be the smallest positive integer not belonging to the set $\{x_{i}, x_{i}+ik | i = 1, . . . , n\}$. Prove that there is a real number $a$ such that $x_{n}= [an]$ for all $n \in\mathbb{ N}$.
2000 Vietnam Team Selection Test
Two players alternately replace the stars in the expression

$$*x^{2000}+*x^{1999}+...+*x+1 $$

by real numbers. The player who makes the last move loses if the resulting polynomial has a real root $t$ with $|t| < 1$, and wins otherwise. Give a winning strategy for one of the players.
2000 Vietnam Team Selection Test
Let $a, b, c$ be pairwise coprime natural numbers. A positive integer $n$ is said to be stubborn if it cannot be written in the form

$n = bcx+cay+abz$, for some $x, y, z \in\mathbb{ N}.$ Determine the number of stubborn numbers.
2000 Vietnam Team Selection Test
Let $a > 1$ and $r > 1$ be real numbers.

(a) Prove that if $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ is a function satisfying the conditions

(i) $f (x)^{2}\leq ax^{r}f (\frac{x}{a})$ for all $x > 0$,

(ii) $f (x) < 2^{2000}$ for all $x < \frac{1}{2^{2000}}$,

then $f (x) \leq x^{r}a^{1-r}$ for all $x > 0$.

(b) Construct a function $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ satisfying condition (i) such that for all $x > 0, f (x) > x^{r}a^{1-r}$ .
2000 Vietnam Team Selection Test
A collection of $2000$ congruent circles is given on the plane such that no

two circles are tangent and each circle meets at least two other circles.

Let $N$ be the number of points that belong to at least two of the circles.

Find the smallest possible value of $N$.
1999 Vietnam Team Selection Test
Let an odd prime $p$ be a given number satisfying $2^h \neq 1 \pmod{p}$ for all $h < p-1, h \in \mathbb{N}^{*},$ and an even integer $a \in \left(\frac{p}{2},p \right).$ Let us consider the sequence $\{a_n\}^{\infty}_{n=0}$ defined by $a_0 = a$ and $a_{n+1} = p - b_n$ for $n = 0, 1, 2, \ldots$, where $b_n$ is the greatest odd divisor of $a_n.$ Show that $\{a_n\}$ is periodical and find its least positive period.
1999 Vietnam Team Selection Test
Two polynomials $f(x)$ and $g(x)$ with real coefficients are called similar if there exist nonzero real number a such that  $f(x) = q \cdot g(x)$ for all $x \in R$.



I. Show that there exists a polynomial $P(x)$ of degree 1999 with real coefficients which satisfies the condition: $(P(x))^2 - 4$ and $(P'(x))^2 \cdot (x^2-4)$ are similar.



II. How many polynomials of degree 1999 are there which have above mentioned property.
1999 Vietnam Team Selection Test
Let a convex polygon $H$ be given. Show that for every real number $a \in (0, 1)$ there exist 6 distinct points on the sides of $H$, denoted by $A_1, A_2, \ldots, A_6$ clockwise, satisfying the conditions:



I. $(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3)$.

II. Lines $A_1A_2, A_5A_4$ are equidistant from $A_6A_3$.



(By  $(AB)$ we denote vector $AB$)
1999 Vietnam Team Selection Test
Let a sequence of positive reals  $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$, let $k_n$ be the least positive integer satisfying:



$$\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.$$



Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.
1999 Vietnam Team Selection Test
Let a triangle $ABC$ inscribed in circle $\Gamma$ be given. Circle $\Theta$ lies in angle $Â$ of triangle and touches sides $AB, AC$ at $M_1, N_1$ and touches internally $\Gamma$ at $P_1$. The points $M_2, N_2, P_2$ and $M_3, N_3, P_3$ are defined similarly to angles $B$ and $C$ respectively. Show that $M_1N_1, M_2N_2$ and $M_3N_3$ intersect each other at their midpoints.
1999 Vietnam Team Selection Test
Let a regular polygon with $p$ vertices be given, where $p$ is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes $p$ peanuts, goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanut for the leader, skips the two next vertices and gives the second peanut to the monkey at the next vertex; skip four next vertices gives the second peanut for the monkey at the next vertex ... after giving the $k$-th peanut, he skips the $2 \cdot k$ next vertices and gives $k+1$-th for the monkey at the next vertex. He does so until all $p$ peanuts are delivered.



I. How many monkeys are there which does not receive peanuts?

II. How many edges of polygon are there which satisfying condition: both two monkey at its vertex received peanut(s)?
1998 Vietnam Team Selection Test
Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that  $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.
1998 Vietnam Team Selection Test
In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$.  The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$. Prove that for each positive integer $n \geq 3$, the equation:  $$ (p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8) $$ is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$, and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$.
1998 Vietnam Team Selection Test
Let $p(1), p(2), \ldots, p(k)$ be all primes smaller than $m$, prove that



$$\sum^{k}_{i=1} \frac{1}{p(i)} + \frac{1}{p(i)^2} > ln(ln(m)).$$
1998 Vietnam Team Selection Test
Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.
1998 Vietnam Team Selection Test
Let $d$ be a positive divisor of $5 + 1998^{1998}$. Prove that $d = 2 \cdot x^2 + 2 \cdot x  \cdot y + 3 \cdot y^2$, where $x, y$ are integers if and only if $d$ is congruent to 3 or 7 $\pmod{20}$.
1998 Vietnam Team Selection Test
In a conference there are $n \geq 10$ people. It is known that:



I. Each person knows at least $\left[\frac{n+2}{3}\right]$ other people.

II. For each pair of person $A$ and $B$ who don't know each other, there exist some people $A(1), A(2), \ldots, A(k)$ such that $A$ knows $A(1)$, $A(i)$ knows $A(i+1)$ and $A(k)$ knows $B$.

III. There doesn't exist a Hamilton path.



Prove that: We can divide those people into 2 groups: $A$ group has a Hamilton cycle, and the other contains of people who don't know each other.
1997 Vietnam Team Selection Test
Let $ ABCD$ be a given tetrahedron, with $ BC = a$, $ CA = b$, $ AB = c$, $ DA = a_1$, $ DB = b_1$, $ DC = c_1$. Prove that there is a unique point $ P$ satisfying

$$ PA^2 + a_1^2 + b^2 + c^2 = PB^2 + b_1^2 + c^2 + a^2 = PC^2 + c_1^2 + a^2 + b^2 = PD^2 + a_1^2 + b_1^2 + c_1^2

$$

and for this point $ P$ we have $ PA^2 + PB^2 + PC^2 + PD^2 \ge 4R^2$, where $ R$ is the circumradius of the tetrahedron $ ABCD$. Find the necessary and sufficient condition so that this inequality is an equality.
1997 Vietnam Team Selection Test
There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:

1) from each town there are exactly $ k$ direct routes to $ k$ other towns;

2) if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.
1997 Vietnam Team Selection Test
Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n = 1, 2, 3, \ldots$) satisfying the following conditions:

1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$;

2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n = \gcd\{a_i + a_k | i + k = n\}$.
1997 Vietnam Team Selection Test
The function $ f : \mathbb{N} \to \mathbb{Z}$ is defined by $ f(0) = 2$, $ f(1) = 503$ and $ f(n + 2) = 503f(n + 1) - 1996f(n)$ for all $ n \in\mathbb{N}$. Let $ s_1$, $ s_2$, $ \ldots$, $ s_k$ be arbitrary integers not smaller than $ k$, and let $ p(s_i)$ be an arbitrary prime divisor of $ f\left(2^{s_i}\right)$, ($ i = 1, 2, \ldots, k$). Prove that, for any positive integer $ t$ ($ t\le k$), we have $ 2^t \Big | \sum_{i = 1}^kp(s_i)$ if and only if $ 2^t | k$.
1997 Vietnam Team Selection Test
Find all pairs of positive real numbers $ (a, b)$ such that for every $ n \in\mathbb{N}^*$ and every real root $ x_n$ of the equation $ 4n^2x = \log_2(2n^2x + 1)$ we always have $ a^{x_n} + b^{x_n} \ge 2 + 3x_n$.
1997 Vietnam Team Selection Test
Let $ n$, $ k$, $ p$ be positive integers with $ 2 \le k \le \frac {n}{p + 1}$. Let $ n$ distinct points on a circle be given. These points are colored blue and red so that exactly $ k$ points are blue and, on each arc determined by two consecutive blue points in clockwise direction, there are at least $ p$ red points. How many such colorings are there?
1996 Vietnam Team Selection Test
In the plane we are given $3 \cdot n$ points ($n>$1) no three collinear, and the distance between any two of them  is $\leq 1$. Prove that we can construct $n$ pairwise disjoint triangles such that: The vertex set of these triangles are exactly the given 3n points and the sum of the  area of these triangles $< 1/2$.
1996 Vietnam Team Selection Test
For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides  $\sum^{\left\lfloor \frac{n - 1}{2}\right\rfloor}_{i=0} \binom{n}{2 \cdot i + 1} 3^i$. Find all $n$ such that $f(n) = 1996.$





old versionFor each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides  $\sum^{n + 1/2}_{i=1} \binom{2 \cdot i + 1}{n}$. Find all $n$ such that $f(n) = 1996.$
1996 Vietnam Team Selection Test
Find the minimum value of the expression:



$$f(a,b,c)= (a+b)^4+(b+c)^4+(c+a)^4 - \frac{4}{7} \cdot (a^4+b^4+c^4).$$
1996 Vietnam Team Selection Test
Given 3 non-collinear points $A,B,C$. For each point $M$ in the plane ($ABC$) let $M_1$ be the point symmetric to $M$ with respect to $AB$, $M_2$ be the point symmetric to $M_1$ with respect to $BC$ and $M'$ be the point symmetric to $M_2$ with respect to $AC$. Find all points $M$ such that $MM'$ obtains its minimum. Let this minimum value be $d$. Prove that $d$ does not depend on the order of the axes of symmetry we chose (we have 3 available axes, that is $BC$, $CA$, $AB$. In the first part the order of axes we chose $AB$, $BC$, $CA$, and the second part of the problem states that the value $d$ doesn't depend on this order).
1996 Vietnam Team Selection Test
There are some people in a meeting; each doesn't know at least 56 others, and for any pair, there exist a third one who knows both of them. Can the number of people be 65?
1996 Vietnam Team Selection Test
Find all reals $a$ such that the sequence $\{x(n)\}$, $n=0,1,2, \ldots$  that satisfy: $x(0)=1996$ and $x_{n+1} = \frac{a}{1+x(n)^2}$ for any natural number $n$ has a limit as n goes to infinity.
1995 Vietnam Team Selection Test
Let be given a triangle $ ABC$ with $ BC = a$, $ CA = b$, $ AB = c$. Six distinct points $ A_1$, $ A_2$, $ B_1$, $ B_2$, $ C_1$, $ C_2$ not coinciding with $ A$, $ B$, $ C$ are chosen so that $ A_1$, $ A_2$ lie on line $ BC$; $ B_1$, $ B_2$ lie on $ CA$ and $ C_1$, $ C_2$ lie on $ AB$. Let $ \alpha$, $ \beta$, $ \gamma$ three real numbers satisfy $ \overrightarrow{A_1A_2} = \frac {\alpha}{a}\overrightarrow{BC}$, $ \overrightarrow{B_1B_2} = \frac {\beta}{b}\overrightarrow{CA}$, $ \overrightarrow{C_1C_2} = \frac {\gamma}{c}\overrightarrow{AB}$. Let $ d_A$, $ d_B$, $ d_C$ be respectively the radical axes of the circumcircles of the pairs of triangles $ AB_1C_1$ and $ AB_2C_2$; $ BC_1A_1$ and $ BC_2A_2$; $ CA_1B_1$ and $ CA_2B_2$. Prove that $ d_A$, $ d_B$ and $ d_C$ are concurrent if and only if $ \alpha a + \beta b + \gamma c \neq 0$.
1995 Vietnam Team Selection Test
Find all integers $ k$ such that for infinitely many integers $ n \ge 3$ the polynomial

$$ P(x) =x^{n+ 1}+ kx^n - 870x^2 + 1945x + 1995$$

can be reduced into two polynomials with integer coefficients.
1995 Vietnam Team Selection Test
Find all integers $ a$, $ b$, $ n$ greater than $ 1$ which satisfy

$$ \left(a^3 + b^3\right)^n = 4(ab)^{1995}

$$
1995 Vietnam Team Selection Test
A graph has $ n$ vertices and $ \frac {1}{2}\left(n^2 - 3n + 4\right)$ edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length $ k$ of a circuit in such a graph.
1995 Vietnam Team Selection Test
For any nonnegative integer $ n$, let $ f(n)$ be the greatest integer such that $ 2^{f(n)} | n + 1$. A pair $ (n, p)$ of nonnegative integers is called nice if $ 2^{f(n)} > p$. Find all triples $ (n, p, q)$ of nonnegative integers such that the pairs $ (n, p)$, $ (p, q)$ and $ (n + p + q, n)$ are all nice.
1995 Vietnam Team Selection Test
Consider the function $ f(x) = \frac {2x^3 - 3}{3x^2 - 1}$.



$ 1.$ Prove that there is a continuous function $ g(x)$ on $ \mathbb{R}$ satisfying $ f(g(x)) = x$ and $ g(x) > x$ for all real $ x$.



$ 2.$ Show that there exists a real number $ a > 1$ such that the sequence $ \{a_n\}$, $ n = 1, 2, \ldots$, defined as follows $ a_0 = a$, $ a_{n + 1} = f(a_n)$, $ \forall n\in\mathbb{N}$ is periodic with the smallest period $ 1995$.
1994 Vietnam Team Selection Test
Given a parallelogram $ABCD$. Let $E$ be a point on the side $BC$ and $F$ be a point on the side $CD$ such that the triangles $ABE$ and $BCF$ have the same area. The diaogonal $BD$ intersects $AE$ at $M$ and intersects $AF$ at $N$. Prove that:



I.  There exists a triangle, three sides of which are equal to $BM, MN, ND$.

II. When $E, F$ vary such that the length of $MN$ decreases, the radius of the circumcircle of the above mentioned triangle also decreases.
1994 Vietnam Team Selection Test
Consider the equation



$$x^2 + y^2 + z^2 + t^2 - N \cdot x \cdot y \cdot z \cdot t - N = 0$$



where $N$ is a given positive integer.



a) Prove that for an infinite number of values of $N$, this equation has positive integral solutions (each such solution consists of four positive integers $x, y, z, t$),



b) Let $N = 4 \cdot k \cdot (8 \cdot m + 7)$ where $k,m$ are no-negative integers. Prove that the considered equation has no positive integral solutions.
1994 Vietnam Team Selection Test
Let $P(x)$ be given a polynomial of degree 4, having 4 positive roots. Prove that the equation

$$(1-4 \cdot x) \cdot \frac{P(x)}{x^2} + (x^2 + 4 \cdot x - 1) \cdot \frac{P'(x)}{x^2} - P''(x) = 0$$

has also 4 positive roots.
1994 Vietnam Team Selection Test
Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$.



I. Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$.

II. Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.
1994 Vietnam Team Selection Test
Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying



$$f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)$$



for all $x$.
1994 Vietnam Team Selection Test
Calculate



$$T = \sum \frac{1}{n_1! \cdot n_2! \cdot \cdots n_{1994}! \cdot (n_2 + 2 \cdot n_3 + 3 \cdot n_4 + \ldots + 1993 \cdot n_{1994})!}$$



where the sum is taken over all 1994-tuples of the numbers $n_1, n_2, \ldots, n_{1994} \in \mathbb{N} \cup \{0\}$ satisfying $n_1 + 2 \cdot n_2 + 3 \cdot n_3 + \ldots + 1994 \cdot n_{1994} = 1994.$
1993 Vietnam Team Selection Test
We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without one cell in corner a $P$-rectangle. We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without two cells in opposite (under center of rectangle) corners a $S$-rectangle. Using some squares of size $2 \times 2$, some $P$-rectangles and some $S$-rectangles, one form one rectangle of size $1993 \times 2000$ (figures don’t overlap each other). Let $s$ denote the sum of numbers of squares and $S$-rectangles used in such tiling. Find the maximal value of $s$.
1993 Vietnam Team Selection Test
A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$. Find all real numbers $q$ such that  the sequence $\{u_n\}$ defined by  $u_n = a_n^q$, $n = 1, 2, 3, \ldots$ has nonzero finite limit when $n$ goes to infinity.



THERE MIGHT BE A TYPO!
1993 Vietnam Team Selection Test
Let's consider the real numbers $x_1, x_2, x_3, x_4$ satisfying the condition



$$ \dfrac{1}{2}\le x_1^2+x_2^2+x_3^2+x_4^2\le 1 $$



Find the maximal and the minimal values of expression:



$$ A = (x_1 - 2 \cdot x_2 + x_3)^2 + (x_2 - 2 \cdot x_3 + x_4)^2 + (x_2 - 2 \cdot x_1)^2 + (x_3 - 2 \cdot x_4)^2  $$
1993 Vietnam Team Selection Test
Let  $H$, $I$, $O$ be the orthocenter, incenter and circumcenter of a triangle. Show that $2 \cdot IO \geq IH$. When does the equality hold ?
1993 Vietnam Team Selection Test
Let an integer $k > 1$ be given. For each integer $n > 1$, we put



$$f(n) = k \cdot n \cdot \left(1-\frac{1}{p_1}\right) \cdot \left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_r}\right)$$



where $p_1, p_2, \ldots, p_r$ are all distinct prime divisors of $n$. Find all values $k$ for which the sequence $\{x_m\}$ defined by $x_0 = a$ and $x_{m+1} = f(x_m), m = 0, 1, 2, 3, \ldots$ is bounded for all integers $a > 1$.
1993 Vietnam Team Selection Test
Let $n$ points $A_1, A_2, \ldots, A_n$, ($n>2$), be considered in the space, where no four points are coplanar. Each pair of points $A_i, A_j$ are connected by an edge. Find the maximal value of $n$ for which we can paint all edges by two colors – blue and red  such that the following three conditions hold:



I. Each edge is painted by exactly one color.

II. For $i = 1, 2, \ldots, n$, the number of blue edges with one end $A_i$ does not exceed 4.

III. For every red edge $A_iA_j$, we can find at least one point $A_k$ ($k \neq i, j$) such that the edges $A_iA_k$ and $A_jA_k$ are blue.
1992 Vietnam Team Selection Test
Let two natural number $n > 1$ and $m$ be given. Find the least positive integer $k$ which has the following property: Among $k$ arbitrary integers $a_1, a_2, \ldots, a_k$ satisfying the condition $a_i - a_j$ ( $1 \leq i < j \leq k$) is not divided by $n$, there exist two numbers $a_p, a_s$ ($p \neq s$) such that  $m + a_p - a_s$ is divided by $n$.
1992 Vietnam Team Selection Test
Let a polynomial $f(x)$ be given with real coefficients and has degree greater or equal than 1. Show that for every real number $c > 0$, there exists a positive integer $n_0$ satisfying the  following condition: if polynomial $P(x)$ of degree greater or equal than $n_0$ with real coefficients and has leading coefficient equal to 1 then the number of integers $x$ for which $|f(P(x))| \leq c$ is not greater than degree of $P(x)$.
1992 Vietnam Team Selection Test
Let $ABC$ a triangle be given  with $BC = a$, $CA = b$, $AB = c$ ($a \neq b \neq c \neq a$). In  plane ($ABC$) take the points $A'$, $B'$, $C'$ such that:



I. The pairs of points $A$ and $A'$, $B$ and $B'$, $C$ and $C'$ either all lie in one side either all lie in different sides under the lines $BC$, $CA$, $AB$ respectively;



II. Triangles $A'BC$, $B'CA$, $C'AB$ are similar isosceles triangles.



Find the value of angle $A'BC$ as function of $a, b, c$ such that lengths $AA', BB', CC'$ are not sides of an triangle. (The word "triangle" must be understood in its ordinary meaning: its vertices are not collinear.)
1992 Vietnam Team Selection Test
In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.
1992 Vietnam Team Selection Test
Find all pair of positive integers $(x, y)$ satisfying the equation



$$x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.$$
1992 Vietnam Team Selection Test
In a scientific conference, all participants can speak in total $2 \cdot n$ languages ($n \geq 2$). Each participant can speak exactly two languages and each pair of two participants can have at most one common language. It is known that for every integer $k$, $1 \leq k \leq n-1$ there are at most $k-1$ languages such that each of these languages is spoken by at most $k$ participants. Show that we can choose a group from $2 \cdot n$ participants which in total can speak $2 \cdot n$ languages and each language is spoken by exactly two participants from this group.
1991 Vietnam Team Selection Test
1.) In the plane let us consider a set $S$ consisting of $n \geq 3$ distinct points satisfying the following three conditions:



I. The distance between any two points $\in S$ is not greater than 1.

II. For every point $A \in S$, there are exactly two “neighbor” points, i.e. two points $X, Y \in S$ for which $AX = AY = 1$.

III.  For arbitrary two points $A, B \in S$, let $A', A''$ be the two neighbors of $A, B', B''$ the two neighbors of $B$, then $A'AA'' = B'BB''$.



Is there such a set $S$ if $n = 1991$? If $n = 2000$ ? Explain your answer.
1991 Vietnam Team Selection Test
For a positive integer $ n>2$, let $ \left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a sequence of $ n$ positive reals which is either non-decreasing (this means, we have $ a_{1}\leq a_{2}\leq \ldots \leq a_{n}$) or non-increasing (this means, we have $ a_{1}\geq a_{2}\geq \ldots \geq a_{n}$), and which satisfies $ a_{1}\neq a_{n}$. Let $ x$ and $ y$ be positive reals satisfying $ \frac{x}{y}\geq \frac{a_{1}-a_{2}}{a_{1}-a_{n}}$. Show that:

$$ \frac{a_{1}}{a_{2}\cdot x+a_{3}\cdot y}+\frac{a_{2}}{a_{3}\cdot x+a_{4}\cdot y}+\ldots+\frac{a_{n-1}}{a_{n}\cdot x+a_{1}\cdot y}+\frac{a_{n}}{a_{1}\cdot x+a_{2}\cdot y}\geq \frac{n}{x+y}. $$
1991 Vietnam Team Selection Test
Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have:



$$x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.$$



Show that this sequence has a finite limit. Determine this limit.
1991 Vietnam Team Selection Test
Let $T$ be an arbitrary tetrahedron satisfying the following conditions:



I. Each its side has length not greater than 1,

II. Each of its faces is a right triangle.



Let  $s(T) = S^2_{ABC} + S^2_{BCD} + S^2_{CDA} + S^2_{DAB}$. Find the maximal possible value of $s(T)$.