Navvye/Olympiad-Problems-CSV huggingface dataset:
2021 Vietnam TST
Given a fixed circle $(O)$ and two fixed points $B, C$ on that circle, let $A$ be a moving point on $(O)$ such that $\triangle ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$ and let $AD, BE, CF$ be the three heights of $\triangle ABC$. In two rays $\overrightarrow{FA}, \overrightarrow{EA}$, we pick respectively $M,N$ such that $FM = CE, EN = BF$. Let $L$ be the intersection of $MN$ and $EF$, and let $G \neq L$ be the second intersection of $(LEN)$ and $(LFM)$.



a) Show that the circle $(MNG)$ always goes through a fixed point.



b) Let $AD$ intersects $(O)$ at $K \neq A$. In the tangent line through $D$ of $(DKI)$, we pick $P,Q$ such that $GP \parallel AB, GQ \parallel AC$. Let $T$ be the center of $(GPQ)$. Show that $GT$ always goes through a fixed point.
2021 Vietnam TST
Let $n \geq 3$ be a positive integers and $p$ be a prime number such that $p > 6^{n-1} - 2^n + 1$. Let $S$ be the set of $n$ positive integers with different residues modulo $p$. Show that there exists a positive integer $c$ such that there are exactly two ordered triples $(x,y,z) \in S^3$ with distinct elements, such that $x-y+z-c$ is divisible by $p$.
2019 Vietnam TST
In a country there are $n\geq 2$ cities. Any two cities has exactly one two-way airway. The government wants to license several airlines to take charge of these airways with such following conditions:



i) Every airway can be licensed to exactly one airline.

ii) By choosing one arbitrary airline, we can move from a city to any other cities, using only flights from this airline.



What is the maximum number of airlines that the government can license to satisfy all of these conditions?
2019 Vietnam TST
For each positive integer $n$, show that the polynomial: $$P_n(x)=\sum _{k=0}^n2^k\binom{2n}{2k}x^k(x-1)^{n-k}$$has $n$ real roots.
2019 Vietnam TST
Given an acute scalene triangle $ABC$ inscribed in circle $(O)$. Let $H$ be its orthocenter and $M$ be the midpoint of $BC$. Let $D$ lie on the opposite rays of $HA$ so that $BC=2DM$. Let $D'$ be the reflection of $D$ through line $BC$ and $X$ be the intersection of $AO$ and $MD$.



a) Show that $AM$ bisects $D'X$.



b) Similarly, we define the points $E,F$ like $D$ and $Y,Z$ like $X$. Let $S$ be the intersection of tangent lines from $B,C$ with respect to $(O)$. Let $G$ be the projection of the midpoint of $AS$ to the line $AO$. Show that there exists a point with the same power to all the circles $(BEY),(CFZ),(SGO)$ and $(O)$.
2019 Vietnam TST
Find all triplets of positive integers $(x, y, z)$ such that $2^x+1=7^y+2^z$.
2019 Vietnam TST
Given a scalene triangle $ABC$ inscribed in the circle $(O)$. Let $(I)$ be its incircle and $BI,CI$ cut $AC,AB$ at $E,F$ respectively. A circle passes through $E$ and touches $OB$ at $B$ cuts $(O)$ again at $M$. Similarly, a circle passes through $F$ and touches $OC$ at $C$ cuts $(O)$ again at $N$. $ME,NF$ cut $(O)$ again at $P,Q$. Let $K$ be the intersection of $EF$ and $BC$ and let $PQ$ cuts $BC$ and $EF$ at $G,H$, respectively. Show that the median correspond to $G$ of the triangle $GHK$ is perpendicular to $IO$.
2019 Vietnam TST
In the real axis, there is bug standing at coordinate $x=1$. Each step, from the position $x=a$, the bug can jump to either $x=a+2$ or $x=\frac{a}{2}$. Show that there are precisely $F_{n+4}-(n+4)$ positions (including the initial position) that the bug can jump to by at most $n$ steps.



Recall that $F_n$ is the $n^{th}$ element of the Fibonacci sequence, defined by $F_0=F_1=1$, $F_{n+1}=F_n+F_{n-1}$ for all $n\geq 1$.
2018 Vietnam Team Selection Test
Let $ABC$ be a acute, non-isosceles triangle. $D,\ E,\ F$ are the midpoints of sides $AB,\ BC,\ AC$, resp. Denote by $(O),\ (O')$ the circumcircle and Euler circle of $ABC$. An arbitrary point $P$ lies inside triangle $DEF$ and $DP,\ EP,\ FP$ intersect $(O')$ at $D',\ E',\ F'$, resp. Point $A'$ is the point such that $D'$ is the midpoint of $AA'$. Points $B',\ C'$ are defined similarly.

a. Prove that if $PO=PO'$ then $O\in(A'B'C')$;

b. Point $A'$ is mirrored by $OD$, its image is $X$. $Y,\ Z$ are created in the same manner. $H$ is the orthocenter of $ABC$ and $XH,\ YH,\ ZH$ intersect $BC, AC, AB$ at $M,\ N,\ L$ resp. Prove that $M,\ N,\ L$ are collinear.
2018 Vietnam Team Selection Test
For every positive integer $m$, a $m\times 2018$ rectangle consists of unit squares (called "cell") is called complete if  the following conditions are met:

i. In each cell is written either a "$0$", a "$1$" or nothing;

ii. For any binary string $S$ with length $2018$, one may choose a row and complete the empty cells so that the numbers in that row, if read from left to right, produce $S$ (In particular, if a row is already full and it produces $S$ in the same manner then this condition ii. is satisfied).

A complete rectangle is called minimal, if we remove any of its rows and then making it no longer complete.



a. Prove that for any positive integer $k\le 2018$ there exists a minimal $2^k\times 2018$ rectangle with exactly $k$ columns containing both $0$ and $1$.

b. A minimal $m\times 2018$ rectangle has exactly $k$ columns containing at least some $0$ or $1$ and the rest of columns are empty. Prove that $m\le 2^k$.
2018 Vietnam Team Selection Test
For every positive integer $n\ge 3$, let $\phi_n$ be the set of all positive integers less than and coprime to $n$. Consider the polynomial:

$$P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.$$

a. Prove that $P_n(x)=(x^{r_n}+1)Q_n(x)$ for some positive integer $r_n$ and polynomial $Q_n(x)\in\mathbb{Z}[x]$ (not necessary non-constant polynomial).

b. Find all $n$ such that $P_n(x)$ is irreducible over $\mathbb{Z}[x]$.
2018 Vietnam Team Selection Test
Let $a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right]$ be a real number. Sequences $(u_n),\  (v_n)$ are defined as follows:

$$u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.$$

a. Prove that

$${{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.$$b. Find all values of $a$ in the equality case.
2018 Vietnam Team Selection Test
In a $m\times n$ square grid, with top-left corner is $A$, there is route along the edges of the grid starting from $A$ and visits all lattice points (called "nodes") exactly once and ending also at $A$.



a. Prove that this route exists if and only if at least one of $m,\ n$ is odd.

b. If such a route exists, then what is the least possible of turning points?



*A turning point is a node that is different from $A$ and if two edges on the route intersect at the node are perpendicular.
2018 Vietnam Team Selection Test
Triangle $ABC$ circumscribed $(O)$ has $A$-excircle $(J)$ that touches $AB,\ BC,\ AC$ at $F,\ D,\ E$, resp.



a. $L$ is the midpoint of $BC$. Circle with diameter $LJ$ cuts $DE,\ DF$ at $K,\ H$. Prove that $(BDK),\ (CDH)$ has an intersecting point on $(J)$.

b. Let $EF\cap BC =\{G\}$ and $GJ$ cuts $AB,\ AC$ at $M,\ N$, resp. $P\in JB$ and $Q\in JC$ such that

$$\angle PAB=\angle QAC=90{}^\circ .$$$PM\cap QN=\{T\}$ and $S$ is the midpoint of the larger $BC$-arc of $(O)$. $(I)$ is the incircle of $ABC$. Prove that $SI\cap AT\in (O)$.
2017 Vietnam Team Selection Test
There are $44$ distinct holes in a line and $2017$ ants. Each ant comes out of a hole and crawls along the line with a constant speed into another hole, then comes in. Let $T$ be the set of moments for which the ant comes in or out of the holes. Given that $|T|\leq 45$ and the speeds of the ants are distinct. Prove that there exists two ants that don't collide.
2017 Vietnam Team Selection Test
For each positive integer $n$, set $x_n=\binom{2n}{n}$.

a. Prove that if $\frac{2017^k}{2}<n<2017^k$ for some positive integer $k$ then $2017$ divides $x_n$.

b. Find all positive integer $h>1$ such that there exists positive integers $N,T$ such that $(x_n)_{n>N}$ is periodic mod $h$ with period $T$.
2017 Vietnam Team Selection Test
Triangle $ABC$ with incircle $(I)$ touches the sides $AB, BC, AC$ at $F, D, E$, res. $I_b, I_c$ are $B$- and $C-$ excenters of $ABC$. $P, Q$ are midpoints of $I_bE, I_cF$. $(PAC)\cap AB=\{ A, R\}$, $(QAB)\cap AC=\{ A,S\}$.

a. Prove that $PR, QS, AI$ are concurrent.

b. $DE, DF$ cut $I_bI_c$ at $K, J$, res. $EJ\cap FK=\{ M\}$. $PE, QF$ cut $(PAC), (QAB)$ at $X, Y$ res. Prove that $BY, CX, AM$ are concurrent.
2017 Vietnam Team Selection Test
Triangle $ABC$ is inscribed in circle $(O)$. $A$ varies on $(O)$ such that $AB>BC$. $M$ is the midpoint of $AC$. The circle with diameter $BM$ intersects $(O)$ at $R$. $RM$ intersects $(O)$ at $Q$ and intersects $BC$ at $P$. The circle with diameter $BP$ intersects $AB, BO$ at $K,S$ in this order.

a. Prove that $SR$ passes through the midpoint of $KP$.

b. Let $N$ be the midpoint of $BC$. The radical axis of circles with diameters $AN, BM$ intersects $SR$ at $E$. Prove that $ME$ always passes through a fixed point.
2017 Vietnam Team Selection Test
Given $2017$ positive real numbers $a_1,a_2,\dots ,a_{2017}$. For each $n>2017$, set $$a_n=\max\{ a_{i_1}a_{i_2}a_{i_3}|i_1+i_2+i_3=n, 1\leq i_1\leq i_2\leq i_3\leq n-1\}.$$Prove that there exists a positive integer $m\leq 2017$ and a positive integer $N>4m$ such that $a_na_{n-4m}=a_{n-2m}^2$ for every $n>N$.
2017 Vietnam Team Selection Test
For each integer $n>0$, a permutation $a_1,a_2,\dots ,a_{2n}$ of $1,2,\dots 2n$ is called beautiful if for every $1\leq i<j \leq 2n$, $a_i+a_{n+i}=2n+1$ and $a_i-a_{i+1}\not \equiv a_j-a_{j+1}$ (mod $2n+1$) (suppose that $a_{2n+1}=a_1$).

a. For $n=6$, point out a beautiful  permutation.

b. Prove that there exists a beautiful permutation for every $n$.
2016 Vietnam Team Selection Test
Find all $a,n\in\mathbb{Z}^+$ ($a>2$) such that each prime divisor of $a^n-1$ is also prime divisor of $a^{3^{2016}}-1$
2016 Vietnam Team Selection Test
Let $A$ be a set contains $2000$ distinct integers and $B$ be a set contains $2016$ distinct integers. $K$ is the numbers of pairs $(m,n)$ satisfying $$ \begin{cases} m\in A, n\in B\\ |m-n|\leq 1000 \end{cases} $$Find the maximum value of $K$
2016 Vietnam Team Selection Test
Let $ABC$ be triangle with circumcircle $(O)$ of fixed $BC$, $AB \ne AC$ and $BC$ not a diameter. Let $I$ be the incenter of the triangle $ABC$ and $D = AI \cap BC, E = BI \cap CA, F = CI \cap AB$. The circle passing through $D$ and tangent to $OA$ cuts for second time $(O)$ at $G$ ($G \ne A$). $GE, GF$ cut $(O)$ also at $M, N$ respectively.

a) Let $H = BM \cap CN$. Prove that $AH$ goes through a fixed point.

b) Suppose $BE, CF$ cut $(O)$ also at $L, K$ respectively and $AH \cap KL = P$. On $EF$ take $Q$ for $QP = QI$. Let $J$ be a point of the circimcircle of triangle $IBC$ so that $IJ \perp IQ$. Prove that the midpoint of $IJ$ belongs to a fixed circle.
2016 Vietnam Team Selection Test
Given an acute triangle $ABC$ satisfying $\angle ACB<\angle ABC<\angle ACB+\dfrac{\angle BAC}{2}$. Let $D$ be a point on $BC$ such that $\angle ADC=\angle ACB+\dfrac{\angle BAC}{2}$. Tangent of circumcircle of $ABC$ at $A$ hits $BC$ at $E$. Bisector of $\angle AEB$ intersects $AD$ and $(ADE)$ at $G$ and $F$ respectively, $DF$ hits $AE$ at $H.$

a) Prove that circle with diameter $AE,DF,GH$ go through one common point.

b) On the exterior bisector of $\angle BAC $ and ray $AC$ given point $K$ and $M$ respectively satisfying $KB=KD=KM$,  On the exterior bisector of $\angle BAC$ and ray $AB$ given point $L$ and $N$ respectively satisfying $LC=LD=LN.$ Circle throughs $M,N$ and midpoint $I$ of $BC$ hits $BC$ at $P$ ($P\neq I$). Prove that $BM,CN,AP$ concurrent.
2016 Vietnam Team Selection Test
Given $n$ numbers $a_1,a_2,...,a_n$ ($n\geq 3$) where $a_i\in\{0,1\}$ for all $i=1,2.,,,.n$. Consider $n$ following $n$-tuples $$ \begin{aligned} S_1 & =(a_1,a_2,...,a_{n-1},a_n)\\ S_2 & =(a_2,a_3,...,a_n,a_1)\\ & \vdots\\ S_n & =(a_n,a_1,...,a_{n-2},a_{n-1}).\end{aligned}$$For each tuple $r=(b_1,b_2,...,b_n)$, let $$ \omega (r)=b_1\cdot 2^{n-1}+b_2\cdot 2^{n-2}+\cdots+b_n. $$Assume that the numbers $\omega (S_1),\omega (S_2),...,\omega (S_n)$ receive exactly $k$ different values.



a) Prove that $k|n$ and $\frac{2^n-1}{2^k-1}|\omega (S_i)\quad\forall i=1,2,...,n.$



b) Let $$ \begin{aligned} M & =\max _{i=\overline{1,n}}\omega (S_i)\\ m & =\min _{i=\overline{1,n}}\omega (S_i). \end{aligned} $$Prove that $$ M-m\geq\frac{(2^n-1)(2^{k-1}-1)}{2^k-1}. $$
2016 Vietnam Team Selection Test
Given $16$ distinct real numbers $\alpha_1,\alpha_2,...,\alpha_{16}$. For each polynomial $P$, denote $$ V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}). $$Prove that there is a monic polynomial $Q$, $\deg Q=8$ satisfying:



i) $V(QP)=0$ for all polynomial $P$ has $\deg P<8$.



ii) $Q$ has $8$ real roots (including multiplicity).
2015 Vietnam Team selection test
Let $\alpha$ be the positive root of the equation $x^2+x=5$. Let $n$ be a positive integer number, and let $c_0,c_1,\ldots,c_n\in \mathbb{N}$ be such that $ c_0+c_1\alpha+c_2\alpha^2+\cdots+c_n\alpha^n=2015. $

a. Prove that $c_0+c_1+c_2+\cdots+c_n\equiv 2 \pmod{3}$.

b. Find the minimum value of the sum $c_0+c_1+c_2+\cdots+c_n$.
2015 Vietnam Team selection test
sorry if this has been posted before .



given a fixed circle $(O)$ and two fixed point $B,C$ on it.point A varies on circle $(O)$. let $I$ be the midpoint of $BC$ and $H$ be the orthocenter of $\triangle ABC$. ray $IH$ meet $(O)$ at $K$ ,$AH$ meet $BC$ at $D$ ,$KD$ meet $(O)$ at $M$ .a line pass $M$ and perpendicular to $BC$ meet $AI$ at $N$.

a) prove that $N$ varies on a fixed circle.

b) a circle pass $N$ and tangent to $AK$ at $A$ cut $AB,AC$ at $P,Q$. let $J$ be the midpoint of $PQ$ .prove that $AJ$ pass through a fixed point.
2015 Vietnam Team selection test
A positive interger number $k$ is called “$t-m$”-property if forall positive interger number $a$, there exists a positive integer number $n$ such that

${{1}^{k}}+{{2}^{k}}+{{3}^{k}}+...+{{n}^{k}} \equiv a (\bmod m).$

a) Find all positive integer numbers $k$ which has $t-20$-property.

b) Find smallest positive integer number $k$ which has $t-{{20}^{15}}$-property.
2015 Vietnam Team selection test
Let $ABC$ be a triangle with an interior  point $P$ such that $\angle APB = \angle APC = \alpha$ and $\alpha > 180^o-\angle BAC$. The circumcircle of triangle $APB$ cuts $AC$ at $E$, the circumcircle of triangle $APC$ cuts $AB$ at $F$. Let $Q$ be the point in the triangle $AEF$ such that $\angle AQE = \angle AQF =\alpha$. Let $D$ be the  symmetric point of $Q$ wrt $EF$. Angle bisector of $\angle EDF$ cuts $AP$ at $T$.

a) Prove that $\angle DET = \angle ABC, \angle DFT = \angle ACB$.

b) Straight line $PA$ cuts straight lines $DE, DF$ at $M, N$ respectively. Denote $I, J$ the incenters of the triangles $PEM, PFN$, and $K$ the circumcenter of the triangle $DIJ$. Straight line $DT$ cut $(K)$ at $H$. Prove that $HK$ passes through the incenter of the triangle $DMN$.
2015 Vietnam Team selection test
Find the smallest positive interger number $n$ such that there exists $n$ real numbers $a_1,a_2,\ldots,a_n$ satisfied three conditions as follow:

a. $a_1+a_2+\cdots+a_n>0$;

b. $a_1^3+a_2^3+\cdots+a_n^3<0$;

c. $a_1^5+a_2^5+\cdots+a_n^5>0$.
2014 Vietnam Team Selection Test
Find all $ f:\mathbb{Z}\rightarrow\mathbb{Z} $ such that

$$ f(2m+f(m)+f(m)f(n))=nf(m)+m $$ $ \forall m,n\in\mathbb{Z} $
2014 Vietnam Team Selection Test
In the Cartesian plane is given a set of points with integer coordinate $$ T=\{ (x;y)\mid x,y\in\mathbb{Z} ; \ |x|,|y|\leq 20 ; \ (x;y)\ne (0;0)\} $$We colour some points of $ T $ such that for each point $ (x;y)\in T $ then either $ (x;y) $ or $ (-x;-y) $ is coloured. Denote $ N $ to be the number of couples $ {(x_1;y_1),(x_2;y_2)} $ such that both $ (x_1;y_1) $ and $ (x_2;y_2) $ are coloured and $ x_1\equiv 2x_2 \pmod {41}, y_1\equiv 2y_2 \pmod {41} $. Find the all possible values of $ N $.
2014 Vietnam Team Selection Test
Let $ABC$ be triangle with $A<B<C$ and inscribed in a circle  $(O)$. On the minor arc $ABC$ of $(O)$ and does not contain point $A$, choose an arbitrary point $D$. Suppose $CD$ meets $AB$ at $E$ and $BD$ meets $AC$ at $F$. Let $O_1$ be the incenter  of triangle $EBD$ touches with $EB,ED$ and tangent to $(O)$. Let $O_2$ be the incenter  of triangle $FCD$, touches with $FC,FD$  and tangent to $(O)$.

a) $M$ is a tangency point of $O_1$ with $BE$ and $N$ is a tangency   point of $O_2$ with $CF$. Prove that the circle with diameter $MN$ has a fixed point.

b) A line through $M$ is parallel to $CE$ meets $AC$ at $P$, a  line through $N$ is parallel to  $BF$ meets $AB$ at  $Q$. Prove that the circumcircles of triangles $(AMP),(ANQ)$ are all tangent to a fixed circle.
2014 Vietnam Team Selection Test
a. Let $ABC$ be a triangle with altitude  $AD$ and  $P$ a variable point on $AD$. Lines $PB$ and  $AC$ intersect each other at $E$, lines  $PC$ and  $AB$ intersect each other at $F.$ Suppose $AEDF$ is a quadrilateral inscribed . Prove that $$\frac{PA}{PD}=(\tan B+\tan C)\cot \frac{A}{2}.$$

b. Let $ABC$ be a triangle with orthocentre  $H$ and $P$ a variable point on $AH$. The line through $C$ perpendicular to $AC$ meets $BP$ at $M$,  The line through $B$ perpendicular to $AB$ meets $CP$ at $N.$ $K$ is the projection of  $A$on $MN$. Prove that $\angle BKC+\angle MAN$ is invariant .
2014 Vietnam Team Selection Test
Find all polynomials $P(x),Q(x)$ which have integer coefficients and satify the following condtion: For the sequence $(x_n )$ defined by $$x_0=2014,x_{2n+1}=P(x_{2n}),x_{2n}=Q(x_{2n-1}) \quad  n\geq 1$$

for every positive integer $m$  is a divisor of some non-zero element of $(x_n  )$
2014 Vietnam Team Selection Test
$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$. A filling of integers is called Dien Bien if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$, the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many Dien Bien fillings are there?

Two fillings are the same if one filling can be transformed to the other filling via a translation.



Click to reveal hidden texttranslation from here
2013 Vietnam Team Selection Test
The $ABCD$ is a cyclic quadrilateral  with no parallel sides inscribed in circle $(O, R)$. Let $E$ be the intersection of two diagonals and the angle bisector of $AEB$ cut the lines $AB, BC, CD, DA$ at $M, N, P, Q$ respectively .

a) Prove that the circles $(AQM), (BMN), (CNP), (DPQ)$ are passing through a point. Call that point $K$.

b) Denote $min \,\{AC, BD\} = m$. Prove that $OK \le \dfrac{2R^2}{\sqrt{4R^2-m^2}}$.
2013 Vietnam Team Selection Test
a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares.

b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.
2013 Vietnam Team Selection Test
Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$. Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously:



i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$;

ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$.



Let $F$ be the set of such functions. For each $f\in F$, let $v(f)$ be the set of values of $f$.



a) Proof that $|F|=\infty$.



b) Proof that for each $f\in F$ then $|v(f)|<\infty$.



c) Find the maximum value of $|v(f)|$ for $f\in F$.
2013 Vietnam Team Selection Test
Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ $$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} $$
2013 Vietnam Team Selection Test
Let $ABC$ be a triangle with  $\angle BAC= 45^o$ . Altitudes $AD, BE, CF$ meet at $H$. $EF$ cuts $BC$ at $P$. $I$ is the midpoint of $BC$, $IF$ cuts $PH$ in $Q$.

a) Prove that  $\angle IQH = \angle AIE$.

b) Let $(K)$ be the circumcircle of triangle $ABC$, $(J)$ be the circumcircle of triangle $KPD$. $CK$ cuts circle $(J)$ at $G$, $IG$ cuts $(J)$ at $M$, $JC$ cuts circle of diameter $BC$ at $N$. Prove that $G, N, M, C$ lie on the same circle.
2013 Vietnam Team Selection Test
A cube with size $10\times 10\times 10$ consists of $1000$ unit cubes, all colored white. $A$ and $B$ play a game on this cube. $A$ chooses some pillars with size $1\times 10\times 10$ such that no two pillars share a vertex or side, and turns all chosen unit cubes to black. $B$ is allowed to choose some unit cubes and ask $A$ their colors. How many unit cubes, at least, that $B$ need to choose so that for any answer from $A$, $B$ can always determine the black unit cubes?
2012 Vietnam Team Selection Test
Consider a circle $(O)$ and two fixed points $B,C$ on $(O)$ such that $BC$ is not the diameter of $(O)$. $A$ is an arbitrary point on $(O)$, distinct from $B,C$. Let $D,J,K$ be the midpoints of $BC,CA,AB$, respectively, $E,M,N$ be the feet of perpendiculars from $A$ to $BC$, $B$ to $DJ$, $C$ to $DK$, respectively. The two tangents at $M,N$ to the circumcircle of triangle $EMN$ meet at $T$. Prove that $T$ is a fixed point (as $A$ moves on $(O)$).
2012 Vietnam Team Selection Test
Consider a $m\times n$ rectangular grid with $m$ rows and $n$ columns. There are water fountains on some of the squares. A water fountain can spray water onto any of it's adjacent squares, or a square in the same column such that there is exactly one square between them. Find the minimum number of fountains such that each square can be sprayed in the case that

a) $m=4$;

b) $m=3$.
2012 Vietnam Team Selection Test
Let $p\ge 17$ be a prime. Prove that $t=3$ is the largest positive integer which satisfies the following condition:

For any integers $a,b,c,d$ such that $abc$ is not divisible by $p$ and $(a+b+c)$ is divisible by $p$, there exists integers $x,y,z$ belonging to the set $\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\}$ such that $ax+by+cz+d$ is divisible by $p$.
2012 Vietnam Team Selection Test
Consider the sequence $(x_n)_{n\ge 1}$ where $x_1=1,x_2=2011$ and $x_{n+2}=4022x_{n+1}-x_n$ for all $n\in\mathbb{N}$. Prove that $\frac{x_{2012}+1}{2012}$ is a perfect square.
2012 Vietnam Team Selection Test
Prove that $c=10\sqrt{24}$ is the largest constant such that if there exist positive numbers $a_1,a_2,\ldots ,a_{17}$ satisfying:

$$\sum_{i=1}^{17}a_i^2=24,\ \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i<c $$

then for every $i,j,k$ such that $1\le 1<j<k\le 17$, we have that $x_i,x_j,x_k$ are sides of a triangle.
2012 Vietnam Team Selection Test
There are $42$ students taking part in the Team Selection Test. It is known that every student knows exactly $20$ other students. Show that we can divide the students into $2$ groups or $21$ groups such that the number of students in each group is equal and every two students in the same group know each other.
2011 Vietnam Team Selection Test
A grasshopper rests on the point $(1,1)$ on the plane. Denote by $O,$ the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point $A$ to $B,$ then the area of $\triangle AOB$ is equal to $\frac 12.$

$(a)$ Find all the positive integral poijnts $(m,n)$ which can be covered by the grasshopper after a finite number of steps, starting from $(1,1).$

$(b)$ If a point $(m,n)$ satisfies the above condition, then show that there exists a certain path for the grasshopper to reach $(m,n)$ from $(1,1)$ such that the number of jumps does not exceed $|m-n|.$
2011 Vietnam Team Selection Test
$A$ is a point lying outside a circle $(O)$. The tangents from $A$ drawn to $(O)$ meet the circle at $B,C.$ Let $P,Q$ be points on the rays $AB, AC$ respectively such that $PQ$ is tangent to $(O).$ The parallel lines drawn through $P,Q$ parallel to $CA, BA,$ respectively meet $BC$ at $E,F,$ respectively.

$(a)$ Show that the straight lines $EQ$ always pass through a fixed point $M,$ and $FP$ always pass through a fixed point $N.$

$(b)$ Show that $PM\cdot QN$ is constant.