Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 Vietnam National Olympiad
Given $m\in\mathbb{Z}^+$. Find all natural numbers $n$ that does not exceed $10^m$ satisfying the following conditions:

i) $3|n.$

ii) The digits of $n$ in decimal representation are in the set $\{2,0,1,5\}$.
2015 Vietnam National Olympiad
Given a circumcircle $(O)$ and two fixed points $B,C$ on $(O)$. $BC$ is not the diameter of $(O)$. A point $A$ varies on $(O)$ such that $ABC$ is an acute triangle. $E,F$ is the foot of the altitude from $B,C$ respectively of $ABC$. $(I)$ is a variable circumcircle going through $E$ and $F$ with center $I$.



a) Assume that $(I)$ touches $BC$ at $D$. Probe that $\frac{DB}{DC}=\sqrt{\frac{\cot B}{\cot C}}$.



b) Assume $(I)$ intersects $BC$ at $M$ and $N$. Let $H$ be the orthocenter and $P,Q$ be the intersections of $(I)$ and $(HBC)$. The circumcircle $(K)$ going through $P,Q$ and touches $(O)$ at $T$ ($T$ is on the same side with $A$ wrt $PQ$). Prove that the interior angle bisector of $\angle{MTN}$ passes through a fixed point.
2015 Vietnam National Olympiad
Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and



${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$



Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.
2015 Vietnam National Olympiad
For $a,n\in\mathbb{Z}^+$, consider the following equation: $$ a^2x+6ay+36z=n\quad (1) $$where $x,y,z\in\mathbb{N}$.



a) Find all $a$ such that for all $n\geq 250$, $(1)$ always has natural roots $(x,y,z)$.



b) Given that $a>1$ and $\gcd (a,6)=1$. Find the greatest value of $n$ in terms of $a$ such that $(1)$ doesn't have natural root $(x,y,z)$.
2015 Vietnam National Olympiad
There are $m$ boys and $n$ girls participate in a duet singing contest ($m,n\geq 2$). At the contest, each section there will be one show. And a show including some boy-girl duets where each boy-girl couple will sing with each other no more than one song and each participant will sing at least one song. Two show $A$ and $B$ are considered different if there is a boy-girl couple sing at show $A$ but not show $B$. The contest will end if and only if every possible shows are performed, and each show is performed exactly one time.



a) A show is called depend on a participant $X$ if we cancel all duets that $X$ perform, then there will be at least one another participant will not sing any song in that show. Prove that among every shows that depend on $X$, the number of shows with odd number of songs equal to the number of shows with even number of songs.



b) Prove that the organizers can arrange the shows in order to make sure that the numbers of songs in two consecutive shows have different parity.
2014 Vietnam National Olympiad
Let  $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by  ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and

$$ \begin{cases}  {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\   x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} $$ for all  $n=1,2,3,\ldots$.

Prove that they are converges and find their limits.
2014 Vietnam National Olympiad
Given the polynomial $P(x)=(x^2-7x+6)^{2n}+13$ where $n$ is a positive integer. Prove that $P(x)$ can't be written as a product of $n+1$ non-constant polynomials with integer coefficients.
2014 Vietnam National Olympiad
Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let $A$ be the number of pairs of adjacent red vertices and $B$ be the number of pairs of adjacent blue vertices.

a) Find all possible values of pair $(A,B).$

b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B=14.$ 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.
2014 Vietnam National Olympiad
Let $ABC$ be an acute triangle, $(O)$ be the circumcircle, and $AB<AC.$ Let $I$ be the midpoint of arc $BC$ (not containing $A$). $K$ lies on $AC,$ $K\ne C$ such that $IK=IC.$ $BK$ intersects $(O)$ at the second point $D,$ $D\ne B$ and intersects $AI$ at $E.$ $DI$ intersects $AC$ at $F.$

a) Prove that $EF=\frac{BC}{2}.$

b) $M$ lies on $DI$ such that $CM$ is parallel to $AD.$ $KM$ intersects $BC$ at $N.$ The circumcircle of triangle $BKN$ intersects $(O)$ at the second point $P.$ Prove that $PK$ passes through the midpoint of segment $AD.$
2014 Vietnam National Olympiad
Given a circle $(O)$ and two fixed points $B,C$ on $(O),$ and an arbitrary point $A$ on $(O)$ such that the triangle $ABC$ is acute. $M$ lies on ray $AB,$ $N$ lies on ray $AC$ such that $MA=MC$ and $NA=NB.$ Let $P$ be the intersection of $(AMN)$ and $(ABC),$ $P\ne A.$ $MN$ intersects $BC$ at $Q.$

a) Prove that $A,P,Q$ are collinear.

b) $D$ is the midpoint of $BC.$ Let $K$ be the intersection of $(M,MA)$ and $(N,NA),$ $K\ne A.$ $d$ is the line passing through $A$ and perpendicular to $AK.$ $E$ is the intersection of $d$ and $BC.$ $(ADE)$ intersects $(O)$ at $F,$ $F\ne A.$ Prove that $AF$ passes through a fixed point.
2014 Vietnam National Olympiad
Find the maximum of

$$P=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}$$

where $x,y,z$ are positive real numbers.
2014 Vietnam National Olympiad
Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.
2013 Vietnam National Olympiad
Solve with full solution:

$$\left\{\begin{matrix}\sqrt{(\sin x)^2+\frac{1}{(\sin x)^2}}+\sqrt{(\cos y)^2+\frac{1}{(\cos y)^2}}=\sqrt\frac{20y}{x+y}

\\\sqrt{(\sin y)^2+\frac{1}{(\sin y)^2}}+\sqrt{(\cos x)^2+\frac{1}{(\cos x)^2}}=\sqrt\frac{20x}{x+y}\end{matrix}\right.  $$
2013 Vietnam National Olympiad
Define a sequence $\{a_n\}$ as: $\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.$



Prove that this sequence has a finite limit as $n\to+\infty$ . Also determine the limit.
2013 Vietnam National Olympiad
Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$. The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$.$J$ is midpoint of $KL$.

a) Prove that $D,I,J$ collinear.

b) $B,C$ are fixed points,$A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant.$IE,IF$ cut $(I)$ again at $M,N$ respectively.$MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.
2013 Vietnam National Olympiad
Write down some numbers $a_1,a_2,\ldots, a_n$ from left to right on a line. Step 1, we write $a_1+a_2$ between $a_1,a_2$; $a_2+a_3$ between $a_2,a_3$, …, $a_{n-1}+a_n$ between $a_{n-1},a_n$, and then we have new sequence $b=(a_1, a_1+a_2,a_2,a_2+a_3,a_3, \ldots, a_{n-1}, a_{n-1}+a_n, a_n)$. Step 2, we do the same thing with  sequence b to have the new sequence c again…. And so on. If we do 2013 steps, count the number of the number 2013 appear on the line if

a)	$n=2$, $a_1=1, a_2=1000$

b)	$n=1000$, $a_i=i, i=1,2\ldots, 1000$



Sorry for my bad English

Moderator says: alternate phrasing here:
2013 Vietnam National Olympiad
Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f(0)=0,f(1)=2013$ and

$$(x-y)(f(f^2(x))-f(f^2(y)))=(f(x)-f(y))(f^2(x)-f^2(y))$$

Note: $f^2(x)=(f(x))^2$
2013 Vietnam National Olympiad
Let $ABC$ be a cute triangle.$(O)$ is circumcircle of $\triangle ABC$.$D$ is on arc $BC$ not containing $A$.Line $\triangle$ moved through $H$($H$ is orthocenter of $\triangle ABC$ cuts circumcircle of $\triangle ABH$,circumcircle $\triangle ACH$ again at $M,N$ respectively.

a.Find $\triangle$ satisfy $S_{AMN}$ max

b.$d_{1},d_{2}$ are the line through $M$ perpendicular to $DB$,the line through $N$ perpendicular to $DC$ respectively.

$d_{1}$ cuts $d_{2}$ at $P$.Prove that $P$ move on a fixed circle.
2013 Vietnam National Olympiad
Find all ordered 6-tuples satisfy following system of modular equation:

$ab+a'b' \equiv 1 $(mod 15)

$bc+b'c' \equiv 1 $(mod 15)

$ca+c'a' \equiv 1 $(mod 15)

Given that $a,b,c,a',b',c' \epsilon (0;1;2;...;14)$
2012 Vietnam National Olympiad
Define a sequence $\{x_n\}$ as: $\left\{\begin{aligned}& x_1=3 \\ & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right.$

Prove that this sequence has a finite limit as $n\to+\infty.$ Also determine the limit.
2012 Vietnam National Olympiad
Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.
2012 Vietnam National Olympiad
Let $ABCD$ be a cyclic quadrilateral with circumcentre $O,$ and the pair of opposite sides not parallel with each other. Let $M=AB\cap CD$ and $N=AD\cap BC.$ Denote, by $P,Q,S,T;$ the intersection of the internal angle bisectors of $\angle MAN$ and $\angle MBN;$ $\angle MBN$ and $\angle MCN;$ $\angle MDN$ and $\angle MAN;$ $\angle MCN$ and $\angle MDN.$ Suppose that the four points $P,Q,S,T$ are distinct.

(a) Show that the four points $P,Q,S,T$ are concyclic. Find the centre of this circle, and denote it as $I.$

(b) Let $E=AC\cap BD.$ Prove that $E,O,I$ are collinear.
2012 Vietnam National Olympiad
Let $n$ be a natural number. There are $n$ boys and $n$ girls standing in a line, in any arbitrary order. A student $X$ will be eligible for receiving $m$ candies, if we can choose two students of opposite sex with $X$ standing on either side of $X$ in $m$ ways. Show that the total number of candies does not exceed $\frac 13n(n^2-1).$
2012 Vietnam National Olympiad
For a group of 5 girls, denoted as $G_1,G_2,G_3,G_4,G_5$ and $12$ boys. There are $17$ chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met:

(a) Each chair has a proper seat.

(b) The order, from left to right, of the girls seating is $G_1; G_2; G_3; G_4; G_5.$

(c) Between $G_1$ and $G_2$ there are at least three boys.

(d) Between $G_4$ and $G_5$ there are at least one boy and most four boys.

How many such arrangements are possible?
2012 Vietnam National Olympiad
Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by

$$v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.$$
2012 Vietnam National Olympiad
Find all $f:\mathbb{R} \to \mathbb{R}$ such that:

(a) For every real number $a$ there exist real number $b$:$f(b)=a$

(b) If $x>y$ then $f(x)>f(y)$

(c) $f(f(x))=f(x)+12x.$
2011 Vietnam National Olympiad
Prove that if $x>0$ and $n\in\mathbb N,$ then we have

$$\frac{x^n(x^{n+1}+1)}{x^n+1}\leq\left(\frac {x+1}{2}\right)^{2n+1}.$$
2011 Vietnam National Olympiad
Let $\langle x_n\rangle$ be a sequence of real numbers defined as

$$x_1=1; x_n=\dfrac{2n}{(n-1)^2}\sum_{i=1}^{n-1}x_i$$

Show that the sequence $y_n=x_{n+1}-x_n$ has finite limits as $n\to \infty.$
2011 Vietnam National Olympiad
Let $AB$ be a diameter of a circle $(O)$ and let $P$ be any point on the tangent drawn at $B$ to $(O).$ Define $AP\cap (O)=C\neq A,$ and let $D$ be the point diametrically opposite to $C.$ If $DP$ meets $(O)$ second time in $E,$ then,



(i) Prove that $AE, BC, PO$ concur at $M.$



(ii) If $R$ is the radius of $(O),$ find  $P$ such that the area of $\triangle AMB$ is maximum, and calculate the area in terms of $R.$
2011 Vietnam National Olympiad
A convex pentagon $ABCDE$ satisfies that the sidelengths and $AC,AD\leq \sqrt 3.$ Let us choose $2011$ distinct points inside this pentagon. Prove that there exists an unit circle with centre on one edge of the pentagon, and which contains at least $403$ points out of the $2011$ given points.

{Edited}

{I posted it correctly before but because of a little confusion deleted the sidelength part, sorry.}
2011 Vietnam National Olympiad
Define the sequence of integers $\langle a_n\rangle$ as;

$$a_0=1, \quad  a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.$$

Prove that $a_{2012}-2010$ is divisible by $2011.$
2011 Vietnam National Olympiad
Let $\triangle ABC$ be a triangle such that $\angle C$ and $\angle B$ are acute. Let $D$ be a variable point on $BC$ such that $D\neq B, C$ and $AD$ is not perpendicular to $BC.$ Let $d$ be the line passing through $D$ and perpendicular to $BC.$ Assume $d \cap AB= E, d \cap AC =F.$ If $M, N, P$ are the incentres of $\triangle AEF, \triangle BDE,\triangle CDF.$ Prove that $A, M, N, P$ are concyclic if and only if $d$ passes through the incentre of $\triangle ABC.$
2011 Vietnam National Olympiad
Let $n\in\mathbb N$ and define $P(x,y)=x^n+xy+y^n.$

Show that we cannot obtain two non-constant polynomials $G(x,y)$ and $H(x,y)$ with real coefficients such that

$P(x,y)=G(x,y)\cdot H(x,y).$
2010 Vietnam National Olympiad
Solve the system equations



$$\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.$$
2010 Vietnam National Olympiad
Let $\{a_{n}\}$ be a sequence which satisfy



$a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$



(a) Find the general fomular for $a_{n}$



(b) Prove that $\{a_{n}\}$ is decreasing sequences
2010 Vietnam National Olympiad
In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$

such that $BC$ not is the diameter.Consider a point $A$ varies in

$(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$

respective is intersect of $BC$ and internal and external bisector

of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through

orthocenter of $\triangle ABC$



and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$.



1/Prove that $MN$ pass through a fixed point



2/Determint the place of $A$ such that $S_{AMN}$ has maxium value
2010 Vietnam National Olympiad
Prove that for each positive integer n,the equation



$x^{2}+15y^{2}=4^{n}$



has at least $n$ integer solution $(x,y)$
2010 Vietnam National Olympiad
Let a positive integer $n$.Consider square table $3*3$.One use $n$

colors to color all cell of table such that



each cell is colored by exactly one color.



Two colored table is same if we can receive them from other by a rotation

through center of $3*3$ table



How many way to color this square table satifies above conditions.
2009 Vietnam National Olympiad
Problem 1.Find all $ (x,y)$ such that:

$$ \{\begin{matrix} \displaystyle\dfrac {1}{\sqrt {1 + 2x^2}} + \dfrac {1}{\sqrt {1 + 2y^2}} & = & \displaystyle\dfrac {2}{\sqrt {1 + 2xy}} \\
\sqrt {x(1 - 2x)} + \sqrt {y(1 - 2y)} & = & \displaystyle\dfrac {2}{9} \end{matrix}\;

$$
2009 Vietnam National Olympiad
The sequence $ \{x_n\}$ is defined by $$ \left\{ \begin{array}{l}x_1  = \frac{1}{2} \\x_n  = \frac{{\sqrt {x_{n - 1} ^2  + 4x_{n - 1} }  + x_{n - 1} }}{2} \\\end{array} \right.$$

Prove that the sequence $ \{y_n\}$, where $ y_n=\sum_{i=1}^{n}\frac{1}{{{x}_{i}}^{2}}$, has a finite limit and find that limit.
2009 Vietnam National Olympiad
Let $ A$, $ B$ be two fixed points and $ C$ is a variable point on the plane such that $ \angle ACB=\alpha$ (constant) ($ 0^{\circ}\le \alpha\le 180^{\circ}$). Let $ D$, $ E$, $ F$ be the projections of the incenter $ I$ of triangle $ ABC$ to its sides $ BC$, $ CA$, $ AB$, respectively. Denoted by $ M$, $ N$ the intersections of $ AI$, $ BI$ with $ EF$, respectively. Prove that the length of the segment $ MN$ is constant and the circumcircle of triangle $ DMN$ always passes through a fixed point.
2009 Vietnam National Olympiad
Let $ a$, $ b$, $ c$ be three real numbers. For each positive integer number $ n$, $ a^n + b^n + c^n$ is an integer number. Prove that there exist three integers $ p$, $ q$, $ r$ such that $ a$, $ b$, $ c$ are the roots of the equation $ x^3 + px^2 + qx + r = 0$.
2009 Vietnam National Olympiad
Let $ S =\{1,2,3, \ldots, 2n\}$ ($ n \in \mathbb{Z}^+$). Ddetermine the number of subsets $ T$ of $ S$ such that there are no 2 element in $ T$ $ a,b$ such that $ |a-b|=\{1,n\}$
2008 Vietnam National Olympiad
Determine the number of solutions of the simultaneous equations $ x^2 + y^3 = 29$ and $ \log_3 x \cdot \log_2 y = 1.$
2008 Vietnam National Olympiad
Given a triangle with acute angle $ \angle BEC,$ let $ E$ be the midpoint of $ AB.$ Point $ M$ is chosen on the opposite ray of $ EC$ such that $ \angle BME = \angle ECA.$ Denote by $ \theta$ the measure of angle $ \angle BEC.$ Evaluate $ \frac{MC}{AB}$ in terms of $ \theta.$
2008 Vietnam National Olympiad
Let $ m = 2007^{2008}$, how many natural numbers n are there such that $ n < m$ and $ n(2n + 1)(5n + 2)$ is divisible by $ m$ (which means that $ m \mid n(2n + 1)(5n + 2)$) ?
2008 Vietnam National Olympiad
he sequence of real number $ (x_n)$ is defined by $ x_1 = 0,$ $ x_2 = 2$ and $ x_{n+2} = 2^{-x_n} + \frac{1}{2}$ $ \forall n = 1,2,3 \ldots$ Prove that the sequence has a limit as $ n$ approaches $ +\infty.$ Determine the limit.
2008 Vietnam National Olympiad
What is the total number of natural numbes divisible by 9 the number of digits of which does not exceed 2008 and at least two of the digits are 9s?
2008 Vietnam National Olympiad
Let $ x, y, z$ be distinct non-negative real numbers. Prove that



$$ \frac{1}{(x-y)^2} + \frac{1}{(y-z)^2} + \frac{1}{(z-x)^2} \geq \frac{4}{xy + yz + zx}.$$



When does the equality hold?
2008 Vietnam National Olympiad
Let $ AD$ is centroid of $ ABC$ triangle. Let $ (d)$ is the perpendicular line with $ AD$. Let $ M$ is a point on $ (d)$. Let $ E, F$ are midpoints of $ MB, MC$ respectively. The line through point $ E$ and perpendicular with $ (d)$ meet $ AB$ at $ P$. The line through point $ F$ and perpendicular with $ (d)$ meet $ AC$ at $ Q$. Let $ (d')$ is a line through point $ M$ and perpendicular with $ PQ$. Prove $ (d')$ always pass a fixed point.
2007 Vietnam National Olympiad
Solve the system of equations:

$\{\begin{array}{l}(1+\frac{12}{3x+y}).\sqrt{x}=2\\(1-\frac{12}{3x+y}).\sqrt{y}=6\end{array}$
2007 Vietnam National Olympiad
Let $x,y$ be integer number with $x,y\neq-1$ so that $\frac{x^{4}-1}{y+1}+\frac{y^{4}-1}{x+1}\in\mathbb{Z}$. Prove that $x^{4}y^{44}-1$ is divisble by $x+1$
2007 Vietnam National Olympiad
Let B,C be fixed points and A be roving point. Let H, G be orthecentre and centroid of triagle ABC. Known midpoint of HG lies on BC, find locus of A
2007 Vietnam National Olympiad
Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.
2007 Vietnam National Olympiad
Given a number $b>0$, find all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that:

$f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}$
2007 Vietnam National Olympiad
Let ABCD be trapezium that is inscribed in circle (O) with larger edge BC. P is a point lying outer segment BC. PA cut (O) at N(that means PA isn't tangent of (O)), the circle with diameter PD intersect (O) at E, DE meet BC at N. Prove that MN always pass through a fixed point.
2006 Vietnam National Olympiad
Solve the following system of equations in real numbers:

$$ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}.  $$
2006 Vietnam National Olympiad
Let $ABCD$ be a convex quadrilateral. Take an arbitrary point $M$ on the line $AB$, and let $N$ be the point of intersection of the circumcircles of triangles $MAC$ and $MBC$ (different from $M$). Prove that:



a) The point $N$ lies on a fixed circle;



b) The line $MN$ passes though a fixed point.
2006 Vietnam National Olympiad
Let $m$, $n$ be two positive integers greater than 3. Consider the table of size $m\times n$ ($m$ rows and $n$ columns)  formed with unit squares. We are putting marbles into unit squares of the table following the instructions:



$-$ each time put 4 marbles into 4 unit squares (1 marble per square) such that the 4 unit squares formes one of the followings 4 pictures (click here to view the pictures).



In each of the following cases, answer with justification to the following question: Is it possible that after a finite number of steps we can set the marbles into all of the unit squares such that the numbers of marbles in each unit square is the same?



a) $m=2004$, $n=2006$;



b) $m=2005$, $n=2006$.
2006 Vietnam National Olympiad
Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$, where $a$, $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that $$ f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} . $$
2006 Vietnam National Olympiad
Find all polynomyals $P(x)$ with real coefficients which satisfy the following equality for all real numbers $x$: $$ P(x^2)+x(3P(x)+P(-x))=(P(x))^2+2x^2 . $$
2006 Vietnam National Olympiad
Let $S$ be a set of 2006 numbers. We call a subset $T$ of $S$ naughty if for any two arbitrary numbers $u$, $v$ (not neccesary distinct) in $T$, $u+v$ is not in $T$. Prove that



1) If $S=\{1,2,\ldots,2006\}$ every naughty subset of $S$ has at most 1003 elements;



2) If $S$ is a set of 2006 arbitrary positive integers, there exists a naughty subset of $S$ which has 669 elements.
2005 Vietnam National Olympiad
Let $x,y$ be real numbers satisfying the condition:

$$x-3\sqrt {x+1}=3\sqrt{y+2} -y$$

Find the greatest value and the smallest value of:

$$P=x+y$$
2005 Vietnam National Olympiad
Let $(O)$ be a fixed circle with the radius $R$. Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ($C\neq A,B$). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$, respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ($D\neq C$). Prove that:

a) $$ CD\leq R  $$

b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$).
2005 Vietnam National Olympiad
Let $A_1A_2A_3A_4A_5A_6A_7A_8$  be convex 8-gon (no three diagonals concruent).

The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of $A_1A_2A_3A_4A_5A_6A_7A_8$ and such convex quadrilaterals will be called "sub quadrilaterals".Find the smallest $n$ satisfying:

We can color n "button" such that for all $i,k \in\{1,2,3,4,5,6,7,8\},i\neq k,s(i,k)$ are the same where $s(i,k)$ denote the number of the  "sub quadrilaterals" has $A_i,A_k$ be the vertices and the intersection of two its diagonals is "button".
2005 Vietnam National Olympiad
Find all function $ f: \mathbb R\to \mathbb R$ satisfying the condition:

$$ f(f(x - y)) = f(x)\cdot f(y) - f(x) + f(y) - xy

$$
2005 Vietnam National Olympiad
Find all triples of natural $ (x,y,n)$ satisfying the condition:

$$ \frac {x! + y!}{n!} = 3^n

$$



Define $ 0! = 1$
2005 Vietnam National Olympiad
Let $\{x_n\}$ be a real sequence defined by:

$$x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n$$

For all $n=1,2,3...$ and a is a real number.

Find all $a$ such that $\{x_n\}$ has finite limit when $n\to +\infty$ and find the finite limit in that cases.
2004 Vietnam National Olympiad
Solve the system of equations $ \begin{cases} x^3 + x(y - z)^2 = 2\\ y^3 + y(z - x)^2 = 30\\ z^3 + z(x - y)^2 = 16\end{cases}$.
2004 Vietnam National Olympiad
In a triangle $ ABC$, the bisector of $ \angle ACB$ cuts the side $ AB$ at $ D$. An arbitrary circle $ (O)$ passing through $ C$ and $ D$ meets the lines $ BC$ and $ AC$ at $ M$ and $ N$ (different from $ C$), respectively.

(a) Prove that there is a circle $ (S)$ touching $ DM$ at $ M$ and $ DN$ at $ N$.

(b) If circle $ (S)$ intersects the lines $ BC$ and $ CA$ again at $ P$ and $ Q$ respectively, prove that the lengths of the segments $ MP$ and $ NQ$ are constant as $ (O)$ varies.