Navvye/Olympiad-Problems-CSV huggingface dataset:
2004 Vietnam National Olympiad
Let $A$ be the set of the $16$ first positive integers. Find the least positive integer $k$ satisfying the condition: In every $k$-subset of $A$, there exist two distinct $a, b \in A$ such that $a^2 + b^2$ is prime.
2004 Vietnam National Olympiad
The sequence $ (x_n)^{\infty}_{n=1}$ is defined by $ x_1 = 1$ and $ x_{n+1} =\frac{(2 + \cos 2\alpha)x_n - \cos^2\alpha}{(2 - 2 \cos 2\alpha)x_n + 2 - \cos 2\alpha}$, for all $ n \in\mathbb{N}$, where $ \alpha$ is a given real parameter. Find all values of $ \alpha$ for which the sequence $ (y_n)$ given by $ y_n = \sum_{k=1}^{n}\frac{1}{2x_k+1}$ has a finite limit when $ n \to +\infty$ and find that limit.
2004 Vietnam National Olympiad
Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$

Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$
2004 Vietnam National Olympiad
Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$.
2003 Vietnam National Olympiad
Let  $f: \mathbb{R}\to\mathbb{R}$ is a function such that $f( \cot x ) = \cos 2x+\sin 2x$ for all $0 < x < \pi$. Define $g(x) = f(x) f(1-x)$ for $-1 \leq x \leq 1$. Find the maximum and minimum values of $g$ on the closed interval $[-1, 1].$
2003 Vietnam National Olympiad
The circles $ C_{1}$ and $ C_{2}$ touch externally at $ M$ and the radius of $ C_{2}$ is larger than that of $ C_{1}$. $ A$ is any point on $ C_{2}$ which does not lie on the line joining the centers of the circles. $ B$ and $ C$ are points on $ C_{1}$ such that $ AB$ and $ AC$ are tangent to $ C_{1}$. The lines $ BM$, $ CM$ intersect $ C_{2}$ again at $ E$, $ F$ respectively. $ D$ is the intersection of the tangent at $ A$ and the line $ EF$. Show that the locus of $ D$ as $ A$ varies is a straight line.
2003 Vietnam National Olympiad
Let $S_{n}$ be the number of permutations $(a_{1}, a_{2}, ... , a_{n})$ of $(1, 2, ... , n)$ such that $1 \leq |a_{k}-k | \leq 2$ for all  $k$. Show that $\frac{7}{4}S_{n-1}< S_{n}< 2 S_{n-1}$ for $n > 6.$
2003 Vietnam National Olympiad
Find the largest positive integer $n$ such that the following equations have integer solutions in $x, y_{1}, y_{2}, ... , y_{n}$ :

$(x+1)^{2}+y_{1}^{2}= (x+2)^{2}+y_{2}^{2}= ... = (x+n)^{2}+y_{n}^{2}.$
2003 Vietnam National Olympiad
Define $p(x) = 4x^{3}-2x^{2}-15x+9, q(x) = 12x^{3}+6x^{2}-7x+1$. Show that each polynomial has just three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$ the largest root of $q(x)$. Show that $A^{2}+3 B^{2}= 4$.
2003 Vietnam National Olympiad
Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$. Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$.
2002 Vietnam National Olympiad
Solve the equation $ \sqrt{4 - 3\sqrt{10 - 3x}} = x - 2$.
2002 Vietnam National Olympiad
An isosceles triangle $ ABC$ with $ AB = AC$ is given on the plane. A variable circle $ (O)$ with center $ O$ on the line $ BC$ passes through $ A$ and does not touch either of the lines $ AB$ and $ AC$. Let $ M$ and $ N$ be the second points of intersection of $ (O)$ with lines $ AB$ and $ AC$, respectively. Find the locus of the orthocenter of triangle $ AMN$.
2002 Vietnam National Olympiad
Let be given two positive integers $ m$, $ n$ with $ m < 2001$, $ n < 2002$. Let distinct real numbers be written in the cells of a $ 2001 \times 2002$ board (with $ 2001$ rows and $ 2002$ columns). A cell of the board is called bad if the corresponding number is smaller than at least $ m$ numbers in the same column and at least $ n$ numbers in the same row. Let $ s$ denote the total number of bad cells. Find the least possible value of $ s$.
2002 Vietnam National Olympiad
Let $ a$, $ b$, $ c$ be real numbers for which the polynomial $ x^3 + ax^2 + bx + c$ has three real roots. Prove that $$ 12ab + 27c \le 6a^3 + 10\left(a^2 - 2b\right)^{\frac {3}{2}}$$ When does equality occur?
2002 Vietnam National Olympiad
Determine for which $ n$ positive integer the equation: $ a + b + c + d = n \sqrt {abcd}$ has positive integer solutions.
2002 Vietnam National Olympiad
For a positive integer $ n$, consider the equation $ \frac{1}{x-1}+\frac{1}{4x-1}+\cdots+\frac{1}{k^2x-1}+\cdots+\frac{1}{n^2x-1}=\frac{1}{2}$.

(a) Prove that, for every $ n$, this equation has a unique root greater than $ 1$, which is denoted by $ x_n$.

(b) Prove that the limit of sequence $ (x_n)$ is $ 4$ as $ n$ approaches infinity.
2001 Vietnam National Olympiad
A circle center $O$ meets a circle center $O'$ at $A$ and $B.$ The line $TT'$ touches the first circle at $T$ and the second at $T'$. The perpendiculars from $T$ and $T'$ meet the line $OO'$ at $S$ and $S'$. The ray $AS$ meets the first circle again at $R$, and the ray $AS'$ meets the second circle again at $R'$. Show that $R, B$ and $R'$ are collinear.
2001 Vietnam National Olympiad
Let $N = 6^{n}$, where $n$ is a positive integer, and let $M = a^{N}+b^{N}$, where $a$ and $b$ are relatively prime integers greater than $1. M$ has at least two odd divisors greater than $1$ are $p,q$. Find the residue of $p^{N}+q^{N}\mod 6\cdot 12^{n}$.
2001 Vietnam National Olympiad
For real $a, b$ define the sequence $x_{0}, x_{1}, x_{2}, ...$ by $x_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}$. If $b = 1$, show that the sequence converges to a finite limit for all $a$. If $b > 2$, show that the sequence diverges for some $a$.
2001 Vietnam National Olympiad
Find the maximum value of $\frac{1}{x^{2}}+\frac{2}{y^{2}}+\frac{3}{z^{2}}$, where $x, y, z$ are positive reals satisfying $\frac{1}{\sqrt{2}}\leq z <\frac{ \min(x\sqrt{2}, y\sqrt{3})}{2}, x+z\sqrt{3}\geq\sqrt{6}, y\sqrt{3}+z\sqrt{10}\geq 2\sqrt{5}.$
2001 Vietnam National Olympiad
Find all real-valued continuous functions defined on the interval $(-1, 1)$ such that $(1-x^{2}) f(\frac{2x}{1+x^{2}}) = (1+x^{2})^{2}f(x)$ for all $x$.
2001 Vietnam National Olympiad
$(a_{1}, a_{2}, ... , a_{2n})$ is a permutation of $\{1, 2, ... , 2n\}$ such that $|a_{i}-a_{i+1}| \neq |a_{j}-a_{j+1}|$ for $i \neq j$. Show that $a_{1}= a_{2n}+n$ iff $1 \leq a_{2i}\leq n$ for $i = 1, 2, ... n.$
2000 Vietnam National Olympiad
Given a real number $ c > 0$, a sequence $ (x_n)$ of real numbers is defined by $ x_{n + 1} = \sqrt {c - \sqrt {c + x_n}}$ for $ n \ge 0$. Find all values of $ c$ such that for each initial value $ x_0$ in $ (0, c)$, the sequence $ (x_n)$ is defined for all $ n$ and has a finite limit $ \lim x_n$ when $ n\to + \infty$.
2000 Vietnam National Olympiad
Two circles $ (O_1)$ and $ (O_2)$ with respective centers $ O_1$, $ O_2$ are given on a plane. Let $ M_1$, $ M_2$ be points on $ (O_1)$, $ (O_2)$ respectively, and let the lines $ O_1M_1$ and $ O_2M_2$ meet at $ Q$. Starting simultaneously from these positions, the points $ M_1$ and $ M_2$ move clockwise on their own circles with the same angular velocity.

(a) Determine the locus of the midpoint of $ M_1M_2$.

(b) Prove that the circumcircle of $ \triangle M_1QM_2$ passes through a fixed point.
2000 Vietnam National Olympiad
Consider the polynomial $ P(x) = x^3 + 153x^2 - 111x + 38$.

(a) Prove that there are at least nine integers $ a$ in the interval $ [1, 3^{2000}]$ for which $ P(a)$ is divisible by $ 3^{2000}$.

(b) Find the number of integers $ a$ in $ [1, 3^{2000}]$ with the property from (a).
2000 Vietnam National Olympiad
For every integer $ n \ge 3$ and any given angle $ \alpha$ with $ 0 < \alpha < \pi$, let $ P_n(x) = x^n \sin\alpha - x \sin n\alpha + \sin(n - 1)\alpha$.

(a) Prove that there is a unique polynomial of the form $ f(x) = x^2 + ax + b$ which divides $ P_n(x)$ for every $ n \ge 3$.

(b) Prove that there is no polynomial $ g(x) = x + c$ which divides $ P_n(x)$ for every $ n \ge 3$.
2000 Vietnam National Olympiad
Find all integers $ n \ge 3$ such that there are $ n$ points in space, with no three on a line and no four on a circle, such that all the circles pass through three points between them are congruent.
2000 Vietnam National Olympiad
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 - 1) = P(x)P(-x)$. Determine the maximum possible number of real roots of $ P(x)$.
1999 Vietnam National Olympiad
Solve the system of equations:

$ (1+4^{2x-y}).5^{1-2x+y}=1+2^{2x-y+1}$

$ y^3+4x+ln(y^2+2x)+1=0$
1999 Vietnam National Olympiad
let a triangle ABC and A',B',C' be the midpoints of the arcs BC,CA,AB respectively of its circumcircle. A'B',A'C' meets BC at $ A_1,A_2$ respectively. Pairs of point $ (B_1,B_2),(C_1,C_2)$ are similarly defined. Prove that $ A_1A_2 = B_1B_2 = C_1C_2$ if and only if triangle ABC is equilateral.
1999 Vietnam National Olympiad
Let $\{x_{n}\}_{n\ge0}$ and $\{y_{n}\}_{n\ge0}$ be two sequences defined recursively as follows $$x_{0}=1, \; x_{1}=4, \; x_{n+2}=3 x_{n+1}-x_{n},$$ $$y_{0}=1, \; y_{1}=2, \; y_{n+2}=3 y_{n+1}-y_{n}.$$



Prove that ${x_{n}}^{2}-5{y_{n}}^{2}+4=0$ for all non-negative integers.

Suppose that $a$, $b$ are two positive integers such that $a^{2}-5b^{2}+4=0$. Prove that there exists a non-negative integer $k$ such that $a=x_{k}$ and $b=y_{k}$.
1999 Vietnam National Olympiad
Given are three positive real numbers $ a,b,c$ satisfying $ abc + a + c = b$. Find the max value of the expression:

$$ P = \frac {2}{a^2 + 1} - \frac {2}{b^2 + 1} + \frac {3}{c^2 + 1}.$$
1999 Vietnam National Olympiad
$ OA, OB, OC, OD$ are 4 rays in space such that the angle between any two is the same. Show that for a variable ray $ OX,$ the sum of the cosines of the angles $ XOA, XOB, XOC, XOD$ is constant and the sum of the squares of the cosines is also constant.
1999 Vietnam National Olympiad
Let $ S = \{0,1,2,\ldots,1999\}$ and $ T = \{0,1,2,\ldots \}.$ Find all functions $ f: T \mapsto S$ such that



(i) $ f(s) = s \quad \forall s \in S.$

(ii) $ f(m+n) = f(f(m)+f(n)) \quad \forall m,n \in T.$
1998 Vietnam National Olympiad
Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit.
1998 Vietnam National Olympiad
Let be given a tetrahedron whose circumcenter is $O$. Draw diameters $AA_{1},BB_{1},CC_{1},DD_{1}$ of the circumsphere of $ABCD$. Let $A_{0},B_{0},C_{0},D_{0}$ be the centroids of triangle $BCD,CDA,DAB,ABC$. Prove that $A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}$ are concurrent at a point, say, $F$. Prove that the line through $F$ and a midpoint of a side of $ABCD$ is perpendicular to the opposite side.
1998 Vietnam National Olympiad
The sequence $\{a_{n}\}_{n\geq 0}$ is defined by $a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...)$. Find the smallest positive integer $h$ satisfying $1998|a_{n+h}-a_{n}\forall n=0,1,2,...$
1998 Vietnam National Olympiad
Does there exist an infinite sequence $\{x_{n}\}$ of reals satisfying the following conditions

i)$|x_{n}|\leq 0,666$ for all $n=1,2,...$

ii)$|x_{m}-x_{n}|\geq \frac{1}{n(n+1)}+\frac{1}{m(m+1)}$ for all $m\not = n$?
1998 Vietnam National Olympiad
Find minimum value of $F(x,y)=\sqrt{(x+1)^{2}+(y-1)^{2}}+\sqrt{(x-1)^{2}+(y+1)^{2}}+\sqrt{(x+2)^{2}+(y+2)^{2}}$, where $x,y\in\mathbb{R}$.
1998 Vietnam National Olympiad
Find all positive integer $n$ such that there exists a $P\in\mathbb{R}[x]$ satisfying $P(x^{1998}-x^{-1998})=x^{n}-x^{-n}\forall x\in\mathbb{R}-\{0\}$.
1997 Vietnam National Olympiad
Given a circle (O,R). A point P lies inside the circle, OP=d, d<R. We consider quadrilaterals ABCD, inscribed in (O), such that AC is perp to BD at point P. Evaluate the maximum and minimum values of the perimeter of ABCD in terms of R and d.
1997 Vietnam National Olympiad
Let n be an integer which is greater than 1, not divisible by 1997.

Let $ a_m=m+\frac{mn}{1997}$ for all m=1,2,..,1996

$ b_m=m+\frac{1997m}{n}$ for all m=1,2,..,n-1

We arrange the terms of two sequence $ (a_i), (b_j)$ in the ascending order to form a new sequence $ c_1\le c_2\le ...\le c_{1995+n}$

Prove that $ c_{k+1}-c_k<2$ for all k=1,2,...,1994+n
1997 Vietnam National Olympiad
Find the number of functions $ f: \mathbb N\rightarrow\mathbb N$ which satisfying:

(i) $ f(1) = 1$

(ii) $ f(n)f(n + 2) = f^2(n + 1) + 1997$ for every natural numbers n.
1997 Vietnam National Olympiad
Let $ k = \sqrt[3]{3}$.

a, Find all polynomials $ p(x)$ with rationl coefficients whose degree are as least as possible such that $ p(k + k^2) = 3 + k$.

b, Does there exist a polynomial $ p(x)$ with integer coefficients satisfying $ p(k + k^2) = 3 + k$
1997 Vietnam National Olympiad
Prove that for evey positive integer n, there exits a positive integer k such that $ 2^n | 19^k - 97$
1997 Vietnam National Olympiad
In the unit cube, given 75 points, no three of which are collinear. Prove that there exits a triangle whose vertices are among the given points and whose area is not greater than 7/72.
1996 Vietnam National Olympiad
Solve the system of equations:

$ \sqrt {3x}(1 + \frac {1}{x + y}) = 2$

$ \sqrt {7y}(1 - \frac {1}{x + y}) = 4\sqrt {2}$
1996 Vietnam National Olympiad
Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.
1996 Vietnam National Olympiad
Let be given integers k and n such that 1<=k<=n. Find the number of ordered k-tuples (a_1,a_2,...,a_n), where a_1, a_2, ..., a_k are different and in the set {1,2,...,n}, satisfying

1) There exist s, t such that 1<=s<t<=k and a_s>a_t.

2) There exists s such that 1<=s<=k and a_s is not congruent to s mod 2.



P.S. This is the only problem from VMO 1996 I cannot find a solution or I cannot solve. But I'm not good at all in combinatoric...
1996 Vietnam National Olympiad
Find all $ f: \mathbb{N}\to\mathbb{N}$ so that :



$ f(n) + f(n + 1) = f(n + 2)f(n + 3) - 1996$
1996 Vietnam National Olympiad
The triangle ABC has BC=1 and $ \angle BAC = a$. Find the shortest distance between its incenter and its centroid. Denote this shortest distance by $ f(a)$. When a varies in the interval $ (\frac {\pi}{3},\pi)$, find the maximum value of $ f(a)$.
1996 Vietnam National Olympiad
Prove that:$a+b+c+d \geq \frac{2}{3}(ab+bc+ca+ad+ac+bd)$

where $a;b;c;d$ are positive real numbers satisfying $2(ab+bc+cd+da+ac+bd)+abc+bcd+cda+dab=16$
1995 Vietnam National Olympiad
Find all real solutions to $ x^3 - 3x^2 - 8x + 40 - 8\sqrt[4]{4x + 4} = 0$
1995 Vietnam National Olympiad
The sequence (a_n) is defined as follows:

$ a_0=1, a_1=3$

For $ n\ge 2$, $ a_{n+2}=a_{n+1}+9a_n$ if n is even, $ a_{n+2}=9a_{n+1}+5a_n$ if n is odd.

Prove that

1) $ (a_{1995})^2+(a_{1996})^2+...+(a_{2000})^2$ is divisible by 20

2) $ a_{2n+1}$ is not a perfect square for every natural numbers $ n$.
1995 Vietnam National Olympiad
Let a non-equilateral triangle $ ABC$ and $ AD,BE,CF$ are its altitudes. On the rays $ AD,BE,CF,$ respectively, let $ A',B',C'$ such that $ \frac {AA'}{AD} = \frac {BB'}{BE} = \frac {CC'}{CF} = k$. Find all values of $ k$ such that $ \triangle A'B'C'\sim\triangle ABC$ for any non-triangle $ ABC.$
1995 Vietnam National Olympiad
Let a tetrahedron $ ABCD$ and $ A',B',C',D'$ be the circumcenters of triangles $ BCD,CDA,DAB,ABC$ respectively. Denote planes $ (P_A),(P_B),(P_C),(P_D)$ be the planes which pass through $ A,B,C,D$ and perpendicular to $ C'D',D'A',A'B',B'C'$ respectively. Prove that these planes have a common point called $ I.$ If $ P$ is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?
1995 Vietnam National Olympiad
Find all poltnomials $ P(x)$ with real coefficients satisfying: For all $ a>1995$, the number of real roots of $ P(x)=a$ (incuding multiplicity of each root) is greater than 1995, and every roots are greater than 1995.
1995 Vietnam National Olympiad
Given an integer $ n\ge 2$ and a reular 2n-gon. Color all verices of the 2n-gon with n colors such that:



(i) Each vertice is colored by exactly one color.

(ii) Two vertices don't have the same color.



Two ways of coloring, satisfying the conditions above, are called equilavent if one obtained from the other by a rotation whose center is the center of polygon. Find the total number of mutually non-equivalent ways of coloring.



Alternative statement:



In how many ways we can color vertices of an regular 2n-polygon using n different colors such that two adjent vertices are colored by different colors. Two colorings which can be received from each other by rotation are considered as the same.
1994 Vietnam National Olympiad
Find all real solutions to

$$x^{3}+3x-3+\ln{(x^{2}-x+1)}=y,$$

$$y^{3}+3y-3+\ln{(y^{2}-y+1)}=z,$$

$$z^{3}+3z-3+\ln{(z^{2}-z+1)}=x.$$
1994 Vietnam National Olympiad
$ABC$ is a triangle. Reflect each vertex in the opposite side to get the triangle $A'B'C'$. Find a necessary and sufficient condition on $ABC$ for $A'B'C'$ to be equilateral.
1994 Vietnam National Olympiad
Define the sequence $\{x_{n}\}$ by $x_{0}=a\in (0,1)$ and $x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...)$. Show that the sequence converges and find its limit.
1994 Vietnam National Olympiad
There are $n+1$ containers arranged in a circle. One container has $n$ stones, the others are empty. A move is to choose two containers $A$ and $B$, take a stone from $A$ and put it in one of the containers adjacent to $B$, and to take a stone from $B$ and put it in one of the containers adjacent to $A$. We can take $A = B$. For which $n$ is it possible by series of moves to end up with one stone in each container except that which originally held $n$ stones.
1994 Vietnam National Olympiad
$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.
1994 Vietnam National Olympiad
Do there exist polynomials $p(x), q(x), r(x)$ whose coefficients are positive integers such that $p(x) = (x^{2}-3x+3) q(x)$ and $q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)$?
1993 Vietnam National Olympiad
$f : [-\sqrt{1995},\sqrt{1995}] \to\mathbb{R}$ is defined by $f(x) = x(1993+\sqrt{1995-x^{2}})$. Find its maximum and minimum values.
1993 Vietnam National Olympiad
$ABCD$ is a quadrilateral such that $AB$ is not parallel to $CD$, and $BC$ is not parallel to $AD$. Variable points $P, Q, R, S$ are taken on $AB, BC, CD, DA$ respectively so that $PQRS$ is a parallelogram. Find the locus of its center.
1993 Vietnam National Olympiad
Find a function $f(n)$ on the positive integers with positive integer values such that $f( f(n) ) = 1993 n^{1945}$ for all $n$.
1993 Vietnam National Olympiad
The tetrahedron $ABCD$ has its vertices on the fixed sphere $S$. Prove that $AB^{2}+AC^{2}+AD^{2}-BC^{2}-BD^{2}-CD^{2}$ is minimum iff $AB\perp AC,AC\perp AD,AD\perp AB$.
1993 Vietnam National Olympiad
$1993$ points are arranged in a circle. At time $0$ each point is arbitrarily labeled $+1$ or $-1$. At times $n = 1, 2, 3, ...$ the vertices are relabeled. At time $n$ a vertex is given the label $+1$ if its two neighbours had the same label at time $n-1$, and it is given the label $-1$ if its two neighbours had different labels at time $n-1$. Show that for some time $n > 1$ the labeling will be the same as at time $1.$
1993 Vietnam National Olympiad
Define the sequences $a_{0}, a_{1}, a_{2}, ...$ and $b_{0}, b_{1}, b_{2}, ...$ by $a_{0}= 2, b_{0}= 1, a_{n+1}= 2a_{n}b_{n}/(a_{n}+b_{n}), b_{n+1}= \sqrt{a_{n+1}b_{n}}$. Show that the two sequences converge to the same limit, and find the limit.
1992 Vietnam National Olympiad
Let $ABCD$ be a tetrahedron satisfying

i)$\widehat{ACD}+\widehat{BCD}=180^{0}$, and

ii)$\widehat{BAC}+\widehat{CAD}+\widehat{DAB}=\widehat{ABC}+\widehat{CBD}+\widehat{DBA}=180^{0}$.

Find value of $[ABC]+[BCD]+[CDA]+[DAB]$ if we know $AC+CB=k$ and $\widehat{ACB}=\alpha$.
1992 Vietnam National Olympiad
For any positive integer $a$, denote $f(a)=|\{b\in\mathbb{N}| b|a$ $\text{and}$ $b\mod{10}\in\{1,9\}\}|$ and $g(a)=|\{b\in\mathbb{N}| b|a$ $\text{and}$ $b\mod{10}\in\{3,7\}\}|$. Prove that $f(a)\geq g(a)\forall a\in\mathbb{N}$.
1992 Vietnam National Olympiad
Let $a,b,c$ be positive reals and sequences $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ defined by $a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}$ for all $k=0,1,2,...$. Prove that $\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty$.
1992 Vietnam National Olympiad
Let $ 9 < n_{1} < n_{2} < \ldots < n_{s} < 1992$ be positive integers and $$ P(x) = 1 + x^{2} + x^{9} + x^{n_{1}} + \cdots + x^{n_{s}} + x^{1992}.$$ Prove that if $ x_{0}$ is real root of $ P(x)$ then $ x_{0}\leq\frac {1 - \sqrt {5}}{2}$.
1992 Vietnam National Olympiad
Let $H$ be a rectangle with angle between two diagonal $\leq 45^{0}$. Rotation $H$ around the its center with angle $0^{0}\leq x\leq 360^{0}$ we have rectangle $H_{x}$. Find $x$ such that $[H\cap H_{x}]$ minimum, where $[S]$ is area of $S$.
1992 Vietnam National Olympiad
Label the squares of a $1991 \times 1992$ rectangle $(m, n)$ with $1 \leq m \leq 1991$ and $1 \leq n \leq 1992$. We wish to color all the squares red. The first move is to color red the squares $(m, n), (m+1, n+1), (m+2, n+1)$for some $m < 1990, n < 1992$. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?
1991 Vietnam National Olympiad
Find all functions $f: \mathbb{R}\to\mathbb{R}$ satisfying:



$\frac{f(xy)+f(xz)}{2} - f(x)f(yz) \geq \frac{1}{4}$  for all $x,y,z \in \mathbb{R}$
1991 Vietnam National Olympiad
Let $k>1$ be an odd integer. For every positive integer n, let $f(n)$ be the greatest positive integer for which $2^{f(n)}$ divides $k^n-1$. Find $f(n)$ in terms of $k$ and $n$.
1991 Vietnam National Olympiad
Three mutually perpendicular rays $O_x,O_y,O_z$ and three points $A,B,C$ on $O_x,O_y,O_z$, respectively. A variable sphere є through $A, B,C$ meets $O_x,O_y,O_z$ again at $A', B',C'$, respectively. Let $M$ and $M'$ be the centroids of triangles $ABC$ and $A'B'C'$. Find the locus of the midpoint of $MM'$.
1991 Vietnam National Olympiad
$1991$ students sit around a circle and play the following game. Starting from some student $A$ and counting clockwise, each student on turn says a number. The numbers are $1,2,3,1,2,3,...$ A student who says $2$ or $3$ must leave the circle. The game is over when there is only one student left. What position was the remaining student sitting at the beginning of the game?
1991 Vietnam National Olympiad
Let $G$ be centroid and $R$ the circunradius of a triangle $ABC$. The extensions of $GA,GB,GC$ meet the circuncircle again at $D,E,F$. Prove that:



$\frac{3}{R} \leq \frac{1}{GD} + \frac{1}{GE} + \frac{1}{GF} \leq \sqrt{3} \leq \frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$
1991 Vietnam National Olympiad
Prove that:

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

where $ x;y;z$ are real numbers saisfying $ x \geq y \geq z \geq 0$
1990 Vietnam National Olympiad
The sequence $ (x_n)$, $ n\in\mathbb{N}^*$ is defined by $ |x_1|<1$, and for all $ n \ge 1$, $$ x_{n+1} =\frac{-x_n +\sqrt{3-3x_n^2}}{2}$$ (a) Find the necessary and sufficient condition for $ x_1$ so that each $ x_n > 0$.

(b) Is this sequence periodic? And why?
1990 Vietnam National Olympiad
At least $ n - 1$ numbers are removed from the set $\{1, 2, \ldots, 2n - 1\}$ according to the following rules:

(i) If $ a$ is removed, so is $ 2a$;

(ii) If $ a$ and $ b$ are removed, so is $ a + b$.

Find the way of removing numbers such that the sum of the remaining numbers is maximum possible.