Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Vietnam Team Selection Test
Let $n$ be a positive integer $\geq 3.$ There are $n$ real numbers $x_1,x_2,\cdots x_n$ that satisfy:

$$\left\{\begin{aligned}&\ x_1\ge x_2\ge\cdots \ge x_n;\\& \ x_1+x_2+\cdots+x_n=0;\\& \ x_1^2+x_2^2+\cdots+x_n^2=n(n-1).\end{aligned}\right.$$

Find the maximum and minimum value of the sum $S=x_1+x_2.$
2011 Vietnam Team Selection Test
Let $\langle a_n\rangle_{n\ge 0}$ be a sequence of integers satisfying $a_0=1, a_1=3$ and $a_{n+2}=1+\left\lfloor \frac{a_{n+1}^2}{a_n}\right\rfloor \ \ \forall n\ge0.$

Prove that

$a_n\cdot a_{n+2}-a_{n+1}^2=2^n$ for every natural number $n.$
2011 Vietnam Team Selection Test
Find all positive integers $n$ such that $A=2^{n+2}(2^n-1)-8\cdot 3^n +1$ is a perfect square.
2011 Vietnam Team Selection Test
Let $n$ be an integer greater than $1.$ $n$ pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of $n.$ They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies.
2010 Vietnam Team Selection Test
Let $n$ be a positive integer. Let  $T_n$ be a set of positive integers such that:

$${T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}$$

Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$  such that $a\equiv b \pmod{110}$
2010 Vietnam Team Selection Test
Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ  $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively.



a) Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$.



b) Prove that $S$ lies on a fix line.
2010 Vietnam Team Selection Test
We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated.



It requires to tile rectangle of size $2008  \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?
2010 Vietnam Team Selection Test
Let $a,b,c$ be positive integers which satisfy the condition: $16(a+b+c)\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.

Prove that

$$\sum_{cyc} \left( \frac{1}{a+b+\sqrt{2a+2c}} \right)^{3}\leq   \frac{8}{9}$$
2010 Vietnam Team Selection Test
We have $n$ countries. Each country have $m$ persons who live in that country ($n>m>1$). We divide $m \cdot n$ persons into $n$ groups each with $m$ members such that there don't exist two persons in any groups who come from one country.

Prove that one can choose $n$ people into one class such that  they come from different groups and different countries.
2010 Vietnam Team Selection Test
Let $S_n $ be sum of squares of the coefficient of the polynomial $(1+x)^n$. Prove that $S_{2n} +1$ is not divisible by $3.$
2009 Vietnam Team Selection Test
Let an acute triangle $ ABC$ with curcumcircle $ (O)$. Call $ A_1,B_1,C_1$ are foots of perpendicular line from $ A,B,C$ to opposite side. $ A_2,B_2,C_2$ are reflect points of $ A_1,B_1,C_1$ over midpoints of $ BC,CA,AB$ respectively. Circle $ (AB_2C_2),(BC_2A_2),(CA_2B_2)$ cut $ (O)$ at $ A_3,B_3,C_3$ respectively.

Prove that: $ A_1A_3,B_1B_3,C_1C_3$ are concurent.
2009 Vietnam Team Selection Test
Let a polynomial $ P(x) = rx^3 + qx^2 + px + 1$ $ (r > 0)$ such that the equation $ P(x) = 0$ has only one real root. A sequence $ (a_n)$ is defined by $ a_0 = 1, a_1 = - p, a_2 = p^2 - q, a_{n + 3} = - pa_{n + 2} - qa_{n + 1} - ra_n$.

Prove that $ (a_n)$ contains an infinite number of nagetive real numbers.
2009 Vietnam Team Selection Test
Let a, b be positive integers. a, b and a.b are not perfect squares.



Prove that at most one of following equations



$ ax^2 - by^2 = 1$ and $ ax^2 - by^2 = - 1$



has solutions in positive integers.
2009 Vietnam Team Selection Test
Let $ a,b,c$ be positive numbers.Find $ k$ such that:

$ (k + \frac {a}{b + c})(k + \frac {b}{c + a})(k + \frac {c}{a + b}) \ge (k + \frac {1}{2})^3$
2009 Vietnam Team Selection Test
Let a circle $ (O)$ with diameter $ AB$. A point $ M$ move inside $ (O)$. Internal bisector of $ \widehat{AMB}$ cut $ (O)$ at $ N$, external bisector of $ \widehat{AMB}$ cut $ NA,NB$ at $ P,Q$. $ AM,BM$ cut circle with diameter $ NQ,NP$ at $ R,S$.

Prove that: median from $ N$ of triangle $ NRS$ pass over a fix point.
2008 Vietnam Team Selection Test
On the plane, given an angle $ xOy$. $ M$ be a mobile point on ray $ Ox$ and $ N$ a mobile point on ray $ Oy$. Let $ d$ be the external angle bisector of angle $ xOy$ and $ I$ be the intersection of $ d$ with the perpendicular bisector of $ MN$. Let $ P$, $ Q$ be two points lie on $ d$ such that $ IP = IQ = IM = IN$, and let $ K$ the intersection of $ MQ$ and $ NP$.



$ 1.$ Prove that $ K$ always lie on a fixed line.



$ 2.$ Let $ d_1$ line perpendicular to $ IM$ at $ M$ and $ d_2$ line perpendicular to $ IN$ at $ N$. Assume that there exist the intersections $ E$, $ F$ of $ d_1$, $ d_2$ from $ d$. Prove that $ EN$, $ FM$ and $ OK$ are concurrent.
2008 Vietnam Team Selection Test
Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$, we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$.
2008 Vietnam Team Selection Test
Let an integer $ n > 3$. Denote the set $ T=\{1,2, \ldots,n\}.$ A subset S of T is called wanting set if S has the property: There exists a positive integer $ c$ which is not greater than $ \frac {n}{2}$ such that $ |s_1 - s_2|\ne c$ for every pairs of arbitrary elements $ s_1,s_2\in S$. How many does a wanting set have at most are there ?
2008 Vietnam Team Selection Test
Let $ m$ and $ n$ be positive integers. Prove that $ 6m | (2m + 3)^n + 1$ if and only if $ 4m | 3^n + 1$
2008 Vietnam Team Selection Test
Let $ k$ be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that $ \frac {AL}{AD} = \frac {BM}{BE} = \frac {CN}{CF} = k$. Denote $ (O_1),(O_2),(O_3)$ be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C.

1) Prove that when $ k = \frac{1}{2}$, three circles $ (O_1),(O_2),(O_3)$ have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles.

2) Find all values of k such that three circles $ (O_1),(O_2),(O_3)$ have exactly two common points
2008 Vietnam Team Selection Test
Consider the set $ M = \{1,2, \ldots ,2008\}$. Paint every number in the set $ M$ with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets:



$ S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}$;

$ S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}$.



Prove that $ 2|S_1| > |S_2|$ (where $ |X|$ denotes the number of elements in a set $ X$).
2007 Vietnam Team Selection Test
Given two sets $A, B$ of positive real numbers such that: $|A| = |B| =n$; $A \neq B$ and $S(A)=S(B)$, where $|X|$ is the number of elements and $S(X)$ is the sum of all elements in set $X$. Prove that we can fill in each unit square of a $n\times n$ square with positive numbers and some zeros such that:



a) the set of the sum of all numbers in each row equals $A$;



b) the set of the sum of all numbers in each column equals $A$.



c) there are at least $(n-1)^{2}+k$ zero numbers in the $n\times n$ array with $k=|A \cap B|$.
2007 Vietnam Team Selection Test
Let $ABC$ be an acute triangle with incricle $(I)$. $(K_{A})$ is the cricle such that $A\in (K_{A})$ and $AK_{A}\perp BC$ and it in-tangent for $(I)$ at $A_{1}$, similary we have $B_{1},C_{1}$.

a) Prove that $AA_{1},BB_{1},CC_{1}$ are concurrent, called point-concurrent is $P$.

b) Assume circles $(J_{A}),(J_{B}),(J_{C})$ are symmetry for excircles $(I_{A}),(I_{B}),(I_{C})$ across midpoints of $BC,CA,AB$ ,resp. Prove that $P_{P/(J_{A})}=P_{P/(J_{B})}=P_{P/(J_{C})}$.



Note. If $(O;R)$ is a circle and $M$ is a point then $P_{M/(O)}=OM^{2}-R^{2}$.
2007 Vietnam Team Selection Test
Given a triangle $ABC$. Find the minimum of

$$\frac{\cos^{2}\frac{A}{2}\cos^{2}\frac{B}{2}}{\cos^{2}\frac{C}{2}}+\frac{\cos^{2}\frac{B}{2}\cos^{2}\frac{C}{2}}{\cos^{2}\frac{A}{2}}+\frac{\cos^{2}\frac{C}{2}\cos^{2}\frac{A}{2}}{\cos^{2}\frac{B}{2}}. $$
2007 Vietnam Team Selection Test
Find all continuous functions $f: \mathbb{R}\to\mathbb{R}$ such that for all real $x$ we have

$$f(x)=f\left(x^{2}+\frac{x}{3}+\frac{1}{9}\right). $$
2007 Vietnam Team Selection Test
Let $A\subset \{1,2,\ldots,4014\}$, $|A|=2007$, such that $a$ does not divide $b$ for all distinct elements $a,b\in A$. For a set $X$ as above let us denote with $m_{X}$ the smallest element in $X$. Find $\min m_{A}$ (for all $A$ with the above properties).
2007 Vietnam Team Selection Test
Let $A_{1}A_{2}\ldots A_{9}$ be a regular $9-$gon. Let $\{A_{1},A_{2},\ldots,A_{9}\}=S_{1}\cup S_{2}\cup S_{3}$ such that $|S_{1}|=|S_{2}|=|S_{3}|=3$. Prove that there exists $A,B\in S_{1}$, $C,D\in S_{2}$, $E,F\in S_{3}$ such that $AB=CD=EF$ and $A \neq B$, $C\neq D$, $E\neq F$.
2006 Vietnam Team Selection Test
Given an acute angles triangle $ABC$, and $H$ is its orthocentre. The external bisector of the angle $\angle BHC$ meets the sides $AB$ and $AC$ at the points $D$ and $E$ respectively. The internal bisector of the angle $\angle BAC$ meets the circumcircle of the triangle $ADE$ again at the point $K$. Prove that $HK$ is through the midpoint of the side $BC$.
2006 Vietnam Team Selection Test
Find all pair of integer numbers $(n,k)$ such that $n$ is not negative and $k$ is greater than $1$, and satisfying that the number:

$$ A=17^{2006n}+4.17^{2n}+7.19^{5n}  $$

can be represented as the product of $k$ consecutive positive integers.
2006 Vietnam Team Selection Test
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.

A natural number $m$ is called good if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.

Find the minimum value of a good number $m$.
2006 Vietnam Team Selection Test
Prove that for all real numbers $x,y,z \in [1,2]$ the following inequality always holds:

$$ (x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 6(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}).  $$

When does the equality occur?
2006 Vietnam Team Selection Test
Given a non-isoceles triangle $ABC$ inscribes a circle $(O,R)$ (center $O$, radius $R$). Consider a varying line $l$ such that $l\perp OA$ and $l$ always intersects the rays $AB,AC$ and these intersectional points are called $M,N$. Suppose that the lines $BN$ and $CM$ intersect, and if the intersectional point is called $K$ then the lines $AK$ and $BC$ intersect.

$1$, Assume that  $P$ is the intersectional point of $AK$ and $BC$. Show that the circumcircle of the triangle $MNP$ is always through a fixed point.

$2$, Assume that $H$ is the orthocentre of the triangle $AMN$. Denote $BC=a$, and $d$ is the distance between $A$ and the line $HK$. Prove that $d\leq\sqrt{4R^2-a^2}$ and the equality occurs iff the line $l$ is through the intersectional point of two lines $AO$ and $BC$.
2006 Vietnam Team Selection Test
The real sequence $\{a_n|n=0,1,2,3,...\}$ defined $a_0=1$ and

$$ a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ).  $$

Denote

$$ A_n=\frac{3}{3 \cdot a_n^2-1}.  $$

Prove that $A_n$ is a perfect square and it has at least $n$ distinct prime divisors.
2005 Vietnam Team Selection Test
Let $(I),(O)$ be the incircle, and, respectiely, circumcircle of $ABC$. $(I)$ touches $BC,CA,AB$ in $D,E,F$ respectively. We are also given three circles $\omega_a,\omega_b,\omega_c$, tangent to $(I),(O)$ in $D,K$ (for $\omega_a$), $E,M$ (for $\omega_b$), and $F,N$ (for $\omega_c$).



a) Show that $DK,EM,FN$ are concurrent in a point $P$;



b) Show that the orthocenter of $DEF$ lies on $OP$.
2005 Vietnam Team Selection Test
Given $n$ chairs around a circle which are marked with numbers from 1 to $n$ .There are $k$, $k \leq 4 \cdot n$  students sitting on those chairs .Two students are called neighbours if there is no student sitting between them.  Between two neighbours students ,there are at less 3 chairs. Find the number of choices of $k$ chairs so that $k$ students can sit on those and the condition is satisfied.
2005 Vietnam Team Selection Test
Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$
2005 Vietnam Team Selection Test
Let be given positive reals $a$, $b$, $c$. Prove that: $\frac{a^{3}}{\left(a+b\right)^{3}}+\frac{b^{3}}{\left(b+c\right)^{3}}+\frac{c^{3}}{\left(c+a\right)^{3}}\geq \frac{3}{8}$.
2005 Vietnam Team Selection Test
Let $p\in \mathbb P,p>3$. Calcute:



a)$S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$  if $ p\equiv 1 \mod 4$



b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$   if $p\equiv 1 \mod 8$
2005 Vietnam Team Selection Test
$n$ is called diamond 2005 if $n=\overline{...ab999...99999cd...}$, e.g. $2005 \times 9$. Let $\{a_n\}:a_n< C\cdot n,\{a_n\}$ is increasing. Prove that $\{a_n\}$ contain infinite diamond 2005.



Compare with this problem.
2004 Vietnam Team Selection Test
Let us consider a set $S = \{ a_1 < a_2 < \ldots < a_{2004}\}$, satisfying the following properties: $f(a_i) < 2003$ and $f(a_i) = f(a_j) \quad \forall i, j$ from $\{1, 2,\ldots , 2004\}$, where $f(a_i)$ denotes number of elements which are relatively prime with $a_i$. Find the least positive integer $k$ for which in every $k$-subset of $S$, having the above mentioned properties there are two distinct elements with greatest common divisor greater than 1.
2004 Vietnam Team Selection Test
Find all real values of $\alpha$, for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation $$ f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y $$ for all $x, y \in \mathbb{R}$.
2004 Vietnam Team Selection Test
In the plane, there are two circles $\Gamma_1, \Gamma_2$ intersecting each other at two points $A$ and $B$. Tangents of $\Gamma_1$ at $A$ and $B$ meet each other at $K$. Let us consider an arbitrary point $M$ (which is different of $A$ and $B$) on $\Gamma_1$. The line $MA$ meets $\Gamma_2$ again at $P$. The line $MK$ meets $\Gamma_1$ again at $C$. The line $CA$ meets $\Gamma_2 $ again at $Q$. Show that the midpoint of $PQ$ lies on the line $MC$ and the line $PQ$ passes through a fixed point when $M$ moves on $\Gamma_1$.



[Moderator edit: This problem was also discussed on  .]
2004 Vietnam Team Selection Test
Let $ \left\{x_n\right\}$, with $ n = 1, 2, 3, \ldots$, be a sequence defined by $ x_1 = 603$, $ x_2 = 102$ and $ x_{n + 2} = x_{n + 1} + x_n + 2\sqrt {x_{n + 1} \cdot x_n - 2}$ $ \forall n \geq 1$. Show that:



(1) The number $ x_n$ is a positive integer for every $ n \geq 1$.



(2) There are infinitely many positive integers $ n$ for which the decimal representation of $ x_n$ ends with 2003.



(3) There exists no positive integer $ n$ for which the decimal representation of $ x_n$ ends with 2004.
2004 Vietnam Team Selection Test
Let us consider a convex hexagon ABCDEF. Let $A_1, B_1,C_1, D_1, E_1, F_1$ be midpoints of the sides $AB, BC, CD, DE, EF,FA$ respectively. Denote by $p$ and $p_1$, respectively, the perimeter of the hexagon $ A B C D E F $ and hexagon $ A_1B_1C_1D_1E_1F_1 $. Suppose that all inner angles of hexagon $ A_1B_1C_1D_1E_1F_1 $ are equal. Prove that $$ p \geq \frac{2 \cdot \sqrt{3}}{3} \cdot p_1 .$$ When does equality hold ?
2004 Vietnam Team Selection Test
Let $S$ be the set of positive integers in which the greatest and smallest elements are relatively prime. For natural $n$, let $S_n$ denote the set of natural numbers which can be represented as sum of at most $n$ elements (not necessarily different) from $S$. Let $a$ be greatest element from $S$. Prove that there are positive integer $k$ and integers $b$ such that $|S_n| = a \cdot n + b$ for all $ n > k $.
2003 Vietnam Team Selection Test
Let be four positive integers $m, n, p, q$, with $p < m$ given and $q < n$. Take four points $A(0; 0), B(p; 0), C (m; q)$ and $D(m; n)$ in the coordinate plane. Consider the paths $f$ from $A$ to $D$ and the paths $g$ from $B$ to $C$ such that when going along $f$ or $g$, one goes only in the positive directions of coordinates and one can only change directions (from the positive direction of one axe coordinate into the the positive direction of the other axe coordinate) at the points with integral coordinates. Let $S$ be the number of couples $(f, g)$ such that $f$ and $g$ have no common points. Prove that



$$S = \binom{n}{m+n} \cdot \binom{q}{m+q-p} - \binom{q}{m+q} \cdot \binom{n}{m+n-p}.$$
2003 Vietnam Team Selection Test
Given a triangle $ABC$. Let $O$ be the circumcenter of this triangle $ABC$. Let $H$, $K$, $L$ be the feet of the altitudes of triangle $ABC$ from the vertices $A$, $B$, $C$, respectively. Denote by $A_{0}$, $B_{0}$, $C_{0}$ the midpoints of these altitudes $AH$, $BK$, $CL$, respectively. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$, respectively. Prove that the four lines $A_{0}D$, $B_{0}E$, $C_{0}F$ and $OI$ are concurrent. (When the point $O$ concides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$.)
2003 Vietnam Team Selection Test
Let $f(0, 0) = 5^{2003}, f(0, n) = 0$ for every integer $n \neq 0$ and

$$\begin{array}{c}\ f(m, n) = f(m-1, n) - 2 \cdot \Bigg\lfloor \frac{f(m-1, n)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n-1)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n+1)}{2}\Bigg\rfloor \end{array}$$

for every natural number $m > 0$ and for every integer $n$.



Prove that there exists a positive integer $M$ such that $f(M, n) = 1$ for all integers $n$ such that $|n| \leq \frac{(5^{2003}-1)}{2}$ and $f(M, n) = 0$ for all integers n such that $|n| > \frac{5^{2003}-1}{2}.$
2003 Vietnam Team Selection Test
On the sides of triangle $ABC$ take the points $M_1, N_1, P_1$ such that each line $MM_1, NN_1, PP_1$ divides the perimeter of $ABC$ in two equal parts ($M, N, P$ are respectively the midpoints of the sides $BC, CA, AB$).



I. Prove that the lines $MM_1, NN_1, PP_1$ are concurrent at a point $K$.

II. Prove that among the ratios $\frac{KA}{BC}, \frac{KB}{CA}, \frac{KC}{AB}$ there exist at least a ratio which is not less than $\frac{1}{\sqrt{3}}$.
2003 Vietnam Team Selection Test
Let $A$ be the set of all permutations $a = (a_1, a_2, \ldots, a_{2003})$ of the 2003 first positive integers such that each permutation satisfies the condition: there is no proper subset $S$ of the set $\{1, 2, \ldots, 2003\}$ such that $\{a_k | k \in S\} = S.$



For each $a = (a_1, a_2, \ldots, a_{2003}) \in A$, let $d(a) = \sum^{2003}_{k=1} \left(a_k - k \right)^2.$



I. Find the least value of $d(a)$. Denote this least value by $d_0$.

II. Find all permutations $a \in A$ such that $d(a) = d_0$.
2003 Vietnam Team Selection Test
Let $n$ be a positive integer. Prove that the number $2^n + 1$ has no prime divisor of the form $8 \cdot k - 1$, where $k$ is a positive integer.
2002 Vietnam Team Selection Test
Find all triangles $ABC$ for which $\angle ACB$ is acute and the interior angle bisector of $BC$ intersects the trisectors $(AX, (AY$ of the angle $\angle BAC$ in the points $N,P$ respectively, such that $AB=NP=2DM$, where $D$ is the foot of the altitude from $A$ on $BC$ and $M$ is the midpoint of the side $BC$.
2002 Vietnam Team Selection Test
On a blackboard  a positive integer $n_0$ is written. Two players, $A$ and $B$ are playing a game, which respects  the following rules:



$-$ acting alternatively per turn, each player deletes the number written on the blackboard $n_k$ and writes instead one number denoted with $n_{k+1}$ from the set $\left\{n_k-1, \dsp \left\lfloor\frac {n_k}3\right\rfloor\right\}$;



$-$ player $A$ starts first deleting $n_0$ and replacing it with $n_1\in\left\{n_0-1, \dsp \left\lfloor\frac {n_0}3\right\rfloor\right\}$;



$-$ the game ends when the number on the table is 0 - and the player who wrote it is the winner.



Find which player has a winning strategy in each of the following cases:



a) $n_0=120$;



b) $n_0=\dsp \frac {3^{2002}-1}2$;



c) $n_0=\dsp \frac{3^{2002}+1}2$.
2002 Vietnam Team Selection Test
Let $m$ be a given positive integer which has a prime divisor greater than $\sqrt {2m} +1 $. Find the minimal positive integer $n$ such that there exists a finite set $S$ of distinct positive integers satisfying the following two conditions:



I. $m\leq x\leq n$ for all $x\in S$;



II. the product of all elements in $S$ is the square of an integer.
2002 Vietnam Team Selection Test
Let $n\geq 2$ be an integer and consider an array composed of $n$ rows and $2n$ columns. Half of the elements in the array are colored in red. Prove that for each integer $k$, $1$, there exist $k$ rows such that the array of size $k\times 2n$ formed with these $k$ rows has at least

$$ \frac { k! (n-2k+2) } {(n-k+1)(n-k+2)\cdots (n-1)} $$ columns which contain only red cells.
2002 Vietnam Team Selection Test
Find all polynomials $P(x)$ with integer coefficients such that the polynomial $$ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 $$ is the square of a polynomial with integer coefficients.
2001 Vietnam Team Selection Test
Let a sequence of integers $\{a_n\}$, $n \in \mathbb{N}$ be given, defined by



$$a_0 = 1, a_n= a_{n-1} + a_{[n/3]}$$



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



Show that for all primes $p \leq 13$, there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$.

(Here $[x]$ denotes the integral part of real number $x$).
2001 Vietnam Team Selection Test
In the plane let two circles be given which intersect at two points $A, B$; Let $PT$ be one of the  two common tangent line of these circles ($P, T$ are points of tangency). Tangents at $P$ and $T$ of the circumcircle of triangle $APT$ meet each other at $S$. Let $H$ be a point symmetric to $B$ under $PT$. Show that $A, S, H$ are collinear.
2001 Vietnam Team Selection Test
Some club has 42 members. It’s known that among 31 arbitrary club members, we can find one pair of a boy and a girl that they know each other. Show that from club members we can choose 12 pairs of knowing each other boys and girls.
2001 Vietnam Team Selection Test
Let’s consider the real numbers $a, b, c$ satisfying  the condition



$$21 \cdot a \cdot b + 2 \cdot b \cdot c + 8 \cdot c \cdot a \leq 12.$$



Find the minimal value of the expression



$$P(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$
2001 Vietnam Team Selection Test
Let an integer $n > 1$ be given. In the space with orthogonal coordinate system $Oxyz$ we denote by $T$  the set of all points $(x, y, z)$ with $x, y, z$ are integers, satisfying the condition: $1 \leq x, y, z \leq n$. We  paint all the points of $T$ in such a way that: if the point $A(x_0, y_0, z_0)$ is painted then points $B(x_1, y_1, z_1)$ for which $x_1 \leq x_0, y_1 \leq y_0$ and $z_1 \leq z_0$ could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied.