Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Uzbekistan National Olympiad
Prove that $ \forall n\in\mathbb{N}$,$ \exists a,b,c\in$$\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3) $  such that $n=\frac{ab}{c}$.
2011 Uzbekistan National Olympiad
Given an acute triangle $ABC$ with altituties AD and BE. O circumcinter of $ABC$.If o lies on the segment DE then find the value of $sinAsinBcosC$
2011 Uzbekistan National Olympiad
$A$ graph $G$ arises from $G_{1}$ and $G_{2}$ by pasting them along $S$ if $G$ has induced subgraphs $G_{1}$, $G_{2}$ with $G=G_{1}\cup G_{2}$ and $S$ is such that $S=G_{1}\cap G_{2}.$ A is graph is called chordal if it can be constructed recursively by pasting along complete subgraphs, starting from complete subgraphs. For a graph $G(V,E)$ define its Hilbert polynomial $H_{G}(x)$ to be

$H_{G}(x)=1+Vx+Ex^2+c(K_{3})x^3+c(K_{4})x^4+\ldots+c(K_{w(G)})x^{w(G)},$

where $c(K_{i})$ is the number of $i$-cliques in $G$ and $w(G)$ is the clique number of $G$. Prove that $H_{G}(-1)=0$ if and only if $G$ is chordal or a tree.
2011 Uzbekistan National Olympiad
Find the minimum value of

$|x-y|+\sqrt{(x+2)^2+(y-4)^4}$
2011 Uzbekistan National Olympiad
Let triangle ABC with $ AB=c$ $AC=b$ $BC=a$ $R$ circumradius, $p$ half peremetr of $ABC$.

I f $\frac{acosA+bcosB+ccosC}{asinA+bsinB+csinC}=\frac{p}{9R}$  then find all  value of $cosA$.
2011 Uzbekistan National Olympiad
In acute triangle $ABC$ $AD$ is bisector. $O$ is circumcenter, $H$ is orthocenter. If $AD=AC$ and $AC\perp OH$ . Find all of the value of  $\angle ABC$ and $\angle ACB$.
2011 Uzbekistan National Olympiad
Does existes a function $f:N->N$ and for all positeve integer n

$f(f(n)+2011)=f(n)+f(f(n))$
2005 Uzbekistan National Olympiad
Given a,b  c are   lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$.

Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$
2005 Uzbekistan National Olympiad
Solve in integer the equation

$\frac{1}{2}(x+y)(y+z)(x+z)+(x+y+z)^{3}=1-xyz$
2005 Uzbekistan National Olympiad
Find the last five digits of

$1^{100}+2^{100}+3^{100}+...+999999^{100}$
2005 Uzbekistan National Olympiad
Let $ABCD$ is a cyclic. $K,L,M,N$ are midpoints of segments $AB$, $BC$ $CD$ and $DA$. $H_{1},H_{2},H_{3},H_{4}$ are orthocenters of $AKN$ $KBL$ $LCM$ and $MND$. Prove that $H_{1}H_{2}H_{3}H_{4}$ is a paralelogram.
2004 Uzbekistan National Olympiad
Solve the equation:

$[\sqrt x+\sqrt{x+1}]+[\sqrt {4x+2}]=18$
2004 Uzbekistan National Olympiad
Lenth of a right angle triangle sides are posive integer. Prove that double area of the triangle divides 12.
2004 Uzbekistan National Olympiad
Given a sequence {$a_n$} such that $a_1=2$ and for all positive integer $n\geq 2$ $a_{n+1}=\frac{a_n^4+9}{16a_n}$. Prove that $\frac {4}{5}<a_n<\frac {5}{4}$
2004 Uzbekistan National Olympiad
In triangle $ABC$ $CL$ is a bisector($L$ lies $AB$) $I$ is center  incircle of $ABC$. $G$ is intersection medians. If $a=BC, b=AC, c=AB$  and $CL\perp GI$ then prove that $\frac{a+b+c}{3}=\frac{2ab}{a+b}$
2021 Vietnam National Olympiad
Let $(x_n)$ define by $x_1\in \left(0;\dfrac{1}{2}\right)$ and $x_{n+1}=3x_n^2-2nx_n^3$ for all $n\ge 1$.

a) Prove that $(x_n)$ convergence to $0$.



b) For each $n\ge 1$, let $y_n=x_1+2x_2+\cdots+n x_n$. Prove that $(y_n)$ has a limit.
2021 Vietnam National Olympiad
Find all function $f:\mathbb{R}\to \mathbb{R}$ such that

$$f(x)f(y)=f(xy-1)+yf(x)+xf(y)$$for all $x,y \in \mathbb{R}$
2021 Vietnam National Olympiad
Let $\bigtriangleup ABC$ is not an isosceles triangle and is an acute triangle, $AD,BE,CF$ be the altitudes and $H$ is the orthocenter .Let $I$ is the circumcenter of $\bigtriangleup HEF$ and let $K,J$ is the midpoint of $BC,EF$ respectively.Let $HJ$ intersects $(I)$ again at $G$ and $GK$ intersects $(I)$ at $L\neq G$.

a) Prove that $AL$ is perpendicular to $EF$.

b) Let $AL$ intersects $EF$ at $M$, the line $IM$ intersects the circumcircle $\bigtriangleup IEF$ again at $N$, $DN$ intersects $AB,AC$ at $P$ and $Q$ respectively then prove that $PE,QF,AK$ are concurrent.
2021 Vietnam National Olympiad
For an integer $ n \geq 2 $, let $ s (n) $ be the sum of positive integers not exceeding $ n $ and not relatively prime to $ n $.

a) Prove that $ s (n) = \dfrac {n} {2} \left (n + 1- \varphi (n) \right) $, where $ \varphi (n) $ is the number of integers positive cannot exceed $ n $ and are relatively prime to $ n $.

b) Prove that there is no integer $ n \geq 2 $ such that $ s (n) = s (n + 2021) $
2021 Vietnam National Olympiad
Let the polynomial $P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+a_0$ where $1011\leq a_i\leq 2021$ for all $i=0,1,2,...,21.$ Given that $P(x)$ has an integer root and there exists an positive real number$c$ such that $|a_{k+2}-a_k|\leq c$ for all $k=0,1,...,19.$



a) Prove that $P(x)$ has an only integer root.



b) Prove that $$\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\leq 440c^2.$$
2021 Vietnam National Olympiad
A student divides all $30$ marbles into $5$ boxes numbered $1, 2, 3, 4, 5$ (after being divided, there may be a box with no marbles).

a) How many ways are there to divide marbles into boxes (are two different ways if there is a box with a different number of marbles)?

b) After dividing, the student paints those $30$ marbles by a number of colors (each with the same color, one color can be painted for many marbles), so that there are no $2$ marbles in the same box. have the same color and from any $2$ boxes it is impossible to choose $8$ marbles painted in $4$ colors. Prove that for every division, the student must use no less than $10$ colors to paint the marbles.

c) Show a division so that with exactly $10$ colors the student can paint the marbles that satisfy the conditions in question b).
2021 Vietnam National Olympiad
Let $ ABC $ be an inscribed triangle  in circle $ (O) $. Let $ D $ be the intersection of the two tangent lines of $ (O) $ at $ B $ and $ C $. The circle passing through $ A $ and tangent to $ BC $ at $ B $ intersects the median passing $ A $ of the triangle $ ABC $ at $ G $. Lines $ BG, CG $ intersect $ CD, BD $ at $ E, F $ respectively.

a) The line passing through the midpoint of $ BE $ and $ CF $ cuts $ BF, CE $ at $ M, N $ respectively. Prove that the points $ A, D, M, N $ belong to the same circle.

b) Let $ AD, AG $ intersect the circumcircle of the triangles $ DBC, GBC $ at $ H, K $ respectively. The perpendicular bisectors of $ HK, HE$, and $HF $ cut $ BC, CA$, and $AB $ at $ R, P$, and $Q $ respectively. Prove that the points $ R, P$, and $Q $ are collinear.
2020 Vietnam National Olympiad
Let a sequence $(x_n)$ satisfy :$x_1=1$ and $x_{n+1}=x_n+3\sqrt{x_n} + \frac{n}{\sqrt{x_n}}$,$\forall$n$\ge1$

a) Prove lim$\frac{n}{x_n}=0$

b) Find lim$\frac{n^2}{x_n}$
2020 Vietnam National Olympiad
a)Let$a,b,c\in\mathbb{R}$ and $a^2+b^2+c^2=1$.Prove that:

$|a-b|+|b-c|+|c-a|\le2\sqrt{2}$

b) Let $a_1,a_2,..a_{2019}\in\mathbb{R}$ and $\sum_{i=1}^{2019}a_i^2=1$.Find the maximum of:

$S=|a_1-a_2|+|a_2-a_3|+...+|a_{2019}-a_1|$
2020 Vietnam National Olympiad
Let a sequence $(a_n)$ satisfy: $a_1=5,a_2=13$ and $a_{n+1}=5a_n-6a_{n-1},\forall n\ge2$

a) Prove that $(a_n, a_{n+1})=1,\forall n\ge1$

b) Prove that: $2^{k+1}|p-1\forall k\in\mathbb{N}$, if p is a prime factor of $a_{2^k}$
2020 Vietnam National Olympiad
Let a non-isosceles acute triangle ABC with the circumscribed cycle (O) and the orthocenter H. D, E, F are the reflection of O in the lines BC, CA and AB.

a) $H_a$ is the reflection of H in BC, A' is the reflection of A at O and $O_a$ is the center of (BOC). Prove that $H_aD$ and OA' intersect on (O).

b) Let X is a point satisfy AXDA' is a parallelogram. Prove that (AHX), (ABF), (ACE) have a comom point different than A
2020 Vietnam National Olympiad
Let a system of equations:

$\left\{\begin{matrix}x-ay=yz\\y-az=zx\\z-ax=xy\end{matrix}\right.$

a)Find (x,y,z) if a=0

b)Prove that: the system have 5 distinct roots $\forall$a>1,a$\in\mathbb{R}.$
2020 Vietnam National Olympiad
Let a non-isosceles acute triangle ABC with tha attitude AD, BE, CF and the orthocenter H. DE, DF intersect (AD) at M, N respectively. $P\in AB,Q\in AC$ satisfy $NP\perp AB,MQ\perp AC$

a) Prove that EF is the  tangent line of (APQ)

b) Let T be the tangency point of (APQ) with EF,.DT $\cap$ MN={K}. L is the reflection of A in MN. Prove that MN, EF ,(DLK) pass through a piont
2020 Vietnam National Olympiad
Given a positive integer $n>1$. Denote $T$ a set that contains all ordered sets $(x;y;z)$ such that $x,y,z$ are all distinct positive integers and $1\leq x,y,z\leq 2n$.  Also, a set $A$ containing ordered sets $(u;v)$ is called "connected" with $T$ if for every $(x;y;z)\in T$ then $\{(x;y),(x;z),(y;z)\} \cap A \neq \varnothing$.

a) Find the number of elements of set $T$.

b) Prove that there exists a set "connected" with $T$ that has exactly $2n(n-1)$ elements.

c) Prove that every set "connected" with $T$ has at least $2n(n-1)$ elements.
2019 Vietnam National Olympiad
Let $f:\mathbb{R}\to (0;+\infty )$ be a continuous function such that $\underset{x\to -\infty }{\mathop{\lim }}\,f(x)=\underset{x\to +\infty }{\mathop{\lim }}\,f(x)=0.$

a) Prove that $f(x)$ has the maximum value on $\mathbb{R}.$

b) Prove that there exist two sequeneces  $({{x}_{n}}),({{y}_{n}})$ with ${{x}_{n}}<{{y}_{n}},\forall n=1,2,3,...$ such that they have the same limit when $n$ tends to infinity and $f({{x}_{n}})=f({{y}_{n}})$ for all $n.$
2019 Vietnam National Olympiad
Let $({{x}_{n}})$ be an integer sequence such that $0\le {{x}_{0}}<{{x}_{1}}\le 100$ and

$${{x}_{n+2}}=7{{x}_{n+1}}-{{x}_{n}}+280,\text{  }\forall n\ge 0.$$a) Prove that if ${{x}_{0}}=2,{{x}_{1}}=3$ then for each positive integer $n,$ the sum of divisors of the following number is divisible by $24$ $${{x}_{n}}{{x}_{n+1}}+{{x}_{n+1}}{{x}_{n+2}}+{{x}_{n+2}}{{x}_{n+3}}+2018.$$b) Find all pairs of numbers $({{x}_{0}},{{x}_{1}})$ such that ${{x}_{n}}{{x}_{n+1}}+2019$ is a perfect square for infinitely many nonnegative integer numbers $n.$
2019 Vietnam National Olympiad
For each real coefficient polynomial $f(x)={{a}_{0}}+{{a}_{1}}x+\cdots +{{a}_{n}}{{x}^{n}}$, let

$$\Gamma (f(x))=a_{0}^{2}+a_{1}^{2}+\cdots +a_{m}^{2}.$$Let be given polynomial $P(x)=(x+1)(x+2)\ldots (x+2020).$ Prove that there exists at least $2019$ pairwise distinct polynomials ${{Q}_{k}}(x)$ with $1\le k\le {{2}^{2019}}$ and each of it satisfies two following conditions:

i) $\deg {{Q}_{k}}(x)=2020.$

ii) $\Gamma \left( {{Q}_{k}}{{(x)}^{n}} \right)=\Gamma \left( P{{(x)}^{n}} \right)$ for all positive initeger $n$.
2019 Vietnam National Olympiad
Let $ABC$ be triangle with $H$ is the orthocenter and $I$ is incenter. Denote $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ be the points on the rays $AB, AC, BC, CA, CB$, respectively such that $$AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB.$$Suppose that $B_{1}B_{2}$ cuts $C_{1}C_{2}$ at $A'$, $C_{1}C_{2}$ cuts $A_{1}A_{2}$ at $B'$ and $A_{1}A_{2}$ cuts $B_{1}B_{2}$ at $C'$.

a) Prove that area of triangle $A'B'C'$ is smaller than or equal to the area of triangle $ABC$.

b) Let $J$ be circumcenter of triangle $A'B'C'$. $AJ$ cuts $BC$ at $R$, $BJ$ cuts $CA$ at $S$ and $CJ$ cuts $AB$ at $T$. Suppose that $(AST), (BTR), (CRS)$ intersect at $K$. Prove that if triangle $ABC$ is not isosceles then $HIJK$ is a parallelogram.
2019 Vietnam National Olympiad
Consider polynomial $f(x)={{x}^{2}}-\alpha x+1$ with $\alpha \in \mathbb{R}.$

a) For $\alpha =\frac{\sqrt{15}}{2}$, let write $f(x)$ as the quotient of two polynomials with nonnegative coefficients.

b) Find all value of $\alpha $ such that $f(x)$ can be written as the quotient of two polynomials with nonnegative coefficients.
2019 Vietnam National Olympiad
Let $ABC$ be an acute, nonisosceles triangle with inscribe in a circle $(O)$ and has orthocenter $H$. Denote $M,N,P$ as the midpoints of sides $BC,CA,AB$ and $D,E,F$ as the feet of the altitudes from vertices $A,B,C$ of triangle $ABC$. Let $K$ as the reflection of $H$ through $BC$. Two lines $DE,MP$ meet at $X$; two lines $DF,MN$ meet at $Y$.

a) The line $XY$ cut the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.

b) Two lines $KE,KF$ cuts $(O)$ second time at $S,T$. Prove that $BS,CT,XY$ are concurrent.
2019 Vietnam National Olympiad
There are some papers of the size $5\times 5$ with two sides which are divided into unit squares for both sides. One uses $n$ colors to paint each cell on the paper, one cell by one color, such that two cells on the same positions for two sides are painted by the same color. Two painted papers are consider as the same if the color of two corresponding cells are the same. Prove that there are no more than

$$\frac{1}{8}\left( {{n}^{25}}+4{{n}^{15}}+{{n}^{13}}+2{{n}^{7}} \right)$$pairwise distinct papers that painted by this way.
2018 Vietnam National Olympiad
The sequence $(x_n)$ is defined as follows:

$$x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$for all $n\geq 1$.

a. Prove that $(x_n)$ has a finite limit and find that limit.

b. For every $n\geq 1$, prove that

$$n\leq x_1+x_2+\dots +x_n\leq n+1.$$
2018 Vietnam National Olympiad
We have a scalene acute triangle $ABC$ (triangle with no two equal sides) and a point $D$ on side $BC$. Pick a point $E$ on side $AB$ and a point $F$ on side $AC$ such that $\angle DEB=\angle DFC$. Lines $DF,\, DE$ intersect $AB,\, AC$ at points $M,\, N$, respectively. Denote $(I_1),\, (I_2)$ by the circumcircles of triangles $DEM,\, DFN$ in that order. The circle $(J_1)$ touches $(I_1)$ internally at $D$ and touches $AB$ at $K$, circle $(J_2)$ touches $(I_2)$ internally at $D$ and touches $AC$ at $H$. $P$ is the intersection of $(I_1),\, (I_2)$ different from $D$. $Q$ is the intersection of $(J_1),\, (J_2)$ different from $D$.

a. Prove that all points $D,\, P,\, Q$ lie on the same line.

b. The circumcircles of triangles $AEF,\, AHK$ intersect at $A,\, G$. $(AEF)$ also cut $AQ$ at $A,\, L$. Prove that the tangent at $D$ of $(DQG)$ cuts $EF$ at a point on $(DLG)$.
2018 Vietnam National Olympiad
An investor has two rectangular pieces of land of size $120\times 100$.

a. On the first land, she want to build a house with a rectangular base of size $25\times 35$ and nines circular flower pots with diameter $5$ outside the house. Prove that even the flower pots positions are chosen arbitrary on the land, the remaining land is still sufficient to build the desired house.

b. On the second land, she want to construct a polygonal fish pond such that the distance from an arbitrary point on the land, outside the pond, to the nearest pond edge is not over $5$. Prove that the perimeter of the pond is not smaller than $440-20\sqrt{2}$.
2018 Vietnam National Olympiad
On the Cartesian plane the curve $(C)$ has equation $x^2=y^3$. A line $d$ varies on the plane such that $d$ always cut $(C)$ at three distinct points with $x$-coordinates $x_1,\, x_2,\, x_3$.

a. Prove that the following quantity is a constant:

$$\sqrt[3]{\frac{x_1x_2}{x_3^2}}+\sqrt[3]{\frac{x_2x_3}{x_1^2}}+\sqrt[3]{\frac{x_3x_1}{x_2^2}}.$$b. Prove the following inequality:

$$\sqrt[3]{\frac{x_1^2}{x_2x_3}}+\sqrt[3]{\frac{x_2^2}{x_3x_1}}+\sqrt[3]{\frac{x_3^2}{x_3x_1}}<-\frac{15}{4}.$$
2018 Vietnam National Olympiad
For two positive integers $n$ and $d$, let $S_n(d)$ be the set of all ordered $d$-tuples $(x_1,x_2,\dots ,x_d)$ that satisfy all of the following conditions:

i. $x_i\in \{1,2,\dots ,n\}$ for every $i\in\{1,2,\dots ,d\}$;

ii. $x_i\ne x_{i+1}$ for every $i\in\{1,2,\dots ,d-1\}$;

iii. There does not exist $i,j,k,l\in\{1,2,\dots ,d\}$ such that $i<j<k<l$ and $x_i=x_k,\, x_j=x_l$;

a. Compute $|S_3(5)|$

b. Prove that $|S_n(d)|>0$ if and only if $d\leq 2n-1$.
2018 Vietnam National Olympiad
The sequence $(x_n)$ is defined as follows:

$$x_0=2,\, x_1=1,\, x_{n+2}=x_{n+1}+x_n$$for every non-negative integer $n$.

a. For each $n\geq 1$, prove that $x_n$ is a prime number only if $n$ is a prime number or $n$ has no odd prime divisors

b. Find all non-negative pairs of integers $(m,n)$ such that $x_m|x_n$.
2018 Vietnam National Olympiad
Acute scalene triangle $ABC$ has $G$ as its centroid and $O$ as its circumcenter. Let $H_a,\, H_b,\, H_c$ be the projections of $A,\, B,\, C$ on respective opposite sides and $D,\, E,\, F$ be the midpoints of $BC,\, CA,\, AB$ in that order. $\overrightarrow{GH_a},\, \overrightarrow{GH_b},\, \overrightarrow{GH_c}$ intersect $(O)$ at $X,\,Y,\,Z$ respectively.

a. Prove that the circle $(XCE)$ pass through the midpoint of $BH_a$

b. Let $M,\, N,\, P$ be the midpoints of $AX,\, BY,\, CZ$ respectively. Prove that $\overleftrightarrow{DM},\, \overleftrightarrow{EN},\,\overleftrightarrow{FP}$ are concurrent.
2017 Vietnam National Olympiad
Given $a\in\mathbb{R}$ and a sequence $(u_n)$ defined by $$ \begin{cases} u_1=a\\ u_{n+1}=\frac{1}{2}+\sqrt{\frac{2n+3}{n+1}u_n+\frac{1}{4}}\quad\forall n\in\mathbb{N}^* \end{cases} $$

a) Prove that $(u_n)$ is convergent sequence when $a=5$ and find the limit of the sequence in that case



b) Find all $a$ such that the sequence $(u_n)$ is exist and is convergent.
2017 Vietnam National Olympiad
Is there an integer coefficients polynomial $P(x)$ satisfying $$ \begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2}\\ P(1+\sqrt{5})=2+3\sqrt{5}\end{cases} $$
2017 Vietnam National Olympiad
Given an acute, non isoceles triangle $ABC$ and $(O)$ be its circumcircle, $H$ its orthocenter and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. $AH$ intersects $(O)$ at $D$ ($D\ne A$).



a) Let $I$ be the midpoint of $AH$, $EI$ meets $BD$ at $M$ and $FI$ meets $CD$ at $N$. Prove that $MN\perp OH$.



b) The lines $DE$, $DF$ intersect $(O)$ at $P,Q$ respectively ($P\ne D,Q\ne D$). $(AEF)$ meets $(O)$ and $AO$ at $R,S$ respectively ($R\ne A, S\ne A$). Prove that $BP,CQ,RS$ are concurrent.
2017 Vietnam National Olympiad
Given an integer $n>1$ and a $n\times n$ grid $ABCD$ containing $n^2$ unit squares, each unit square is colored by one of three colors: Black, white and gray. A coloring is called symmetry if each unit square has center on diagonal $AC$ is colored by gray and every couple of unit squares which are symmetry by $AC$ should be both colred by black or white. In each gray square, they label a number $0$, in a white square, they will label a positive integer and in a black square, a negative integer. A label will be called $k$-balance (with $k\in\mathbb{Z}^+$) if it satisfies the following requirements:



i) Each pair of unit squares which are symmetry by $AC$ are labelled with the same integer from the closed interval $[-k,k]$



ii) If a row and a column intersectes at a square that is colored by black, then the set of positive integers on that row and the set of positive integers on that column are distinct.If a row and a column intersectes at a square that is colored by white, then the set of negative integers on that row and the set of negative integers on that column are distinct.



a) For $n=5$, find the minimum value of $k$ such that there is a $k$-balance label for the following grid



[asy]

size(4cm);

pair o = (0,0); pair y = (0,5); pair z = (5,5); pair t = (5,0); dot("$A$", y, dir(180)); dot("$B$", z); dot("$C$", t); dot("$D$", o, dir(180));

fill((0,5)--(1,5)--(1,4)--(0,4)--cycle,gray); 

fill((1,4)--(2,4)--(2,3)--(1,3)--cycle,gray); 

fill((2,3)--(3,3)--(3,2)--(2,2)--cycle,gray); 

fill((3,2)--(4,2)--(4,1)--(3,1)--cycle,gray); 

fill((4,1)--(5,1)--(5,0)--(4,0)--cycle,gray); 

fill((0,3)--(1,3)--(1,1)--(0,1)--cycle,black);

fill((2,5)--(4,5)--(4,4)--(2,4)--cycle,black);

fill((2,1)--(3,1)--(3,0)--(2,0)--cycle,black);

fill((2,1)--(3,1)--(3,0)--(2,0)--cycle,black);

fill((4,3)--(5,3)--(5,2)--(4,2)--cycle,black);    

for (int i=0; i<=5; ++i) { draw((0,i)--(5,i)^^(i,0)--(i,5)); }

[/asy]



b) Let $n=2017$. Find the least value of $k$ such that there is always a $k$-balance label for a symmetry coloring.
2017 Vietnam National Olympiad
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying relation :

$$f(xf(y)-f(x))=2f(x)+xy$$$\forall x,y \in \mathbb{R}$
2017 Vietnam National Olympiad
Prove that

a)$\sum_{k=1}^{1008}kC_{2017}^{k}\equiv 0$ (mod $2017^2$ )



b)$\sum_{k=1}^{504}\left ( -1 \right )^kC_{2017}^{k}\equiv 3\left ( 2^{2016}-1 \right )$ (mod $2017^2$ )
2017 Vietnam National Olympiad
Given an acute triangle $ABC$ and $(O)$ be its circumcircle. Let $G$ be the point on arc $BC$ that doesn't contain $O$ of the circumcircle $(I)$ of triangle $OBC$. The circumcircle of $ABG$ intersects $AC$ at $E$ and circumcircle of $ACG$ intersects $AB$ at $F$ ($E\ne A, F\ne A$).



a) Let $K$ be the intersection of $BE$ and $CF$. Prove that $AK,BC,OG$ are concurrent.



b) Let $D$ be a point on arc $BOC$ (arc $BC$ containing $O$) of $(I)$. $GB$ meets $CD$ at $M$ , $GC$ meets $BD$ at $N$. Assume that $MN$ intersects $(O)$ at $P$ nad $Q$. Prove that when $G$ moves on the arc $BC$ that doesn't contain $O$ of $(I)$, the circumcircle $(GPQ)$ always passes through two fixed points.
2016 Vietnam National Olympiad
Solve the system of equations $\begin{cases}6x-y+z^2=3\\ x^2-y^2-2z=-1\quad\quad (x,y,z\in\mathbb{R}.)\\ 6x^2-3y^2-y-2z^2=0\end{cases}$.
2016 Vietnam National Olympiad
a) Let $(a_n)$ be the sequence defined by $a_n=\ln (2n^2+1)-\ln (n^2+n+1)\,\,\forall n\geq 1.$ Prove that the set $\{n\in\mathbb{N}|\,\{a_n\}<\dfrac{1}{2}\}$ is a finite set;

b) Let $(b_n)$ be the sequence defined by $a_n=\ln (2n^2+1)+\ln (n^2+n+1)\,\,\forall n\geq 1$. Prove that the set $\{n\in\mathbb{N}|\,\{b_n\}<\dfrac{1}{2016}\}$ is an infinite set.
2016 Vietnam National Olympiad
Let $ABC$ be an acute triange with $B,C$ fixed. Let $D$ be the midpoint of $BC$ and $E,F$ be the foot of the perpendiculars from $D$ to $AB,AC$, respectively.

a) Let $O$ be the circumcenter of triangle $ABC$ and $M=EF\cap AO, N=EF\cap BC$. Prove that the circumcircle of triangle $AMN$ passes through a fixed point;

b) Assume that tangents of the circumcircle of triangle $AEF$ at $E,F$ are intersecting at $T$. Prove that $T$ is on a fixed line.
2016 Vietnam National Olympiad
Let $m$ and $n$ be positive integers. A people planted two kind of different trees on a plot tabular grid size $ m \times n $ (each square plant one tree.) A plant called inpressive if two conditions following conditions are met simultaneously:

i) The number of trees in each of kind is equal;

ii) In each row the number of tree of each kind is diffrenent not less than a half of number of cells on that row and In each colum the number of tree of each kind is diffrenent not less than a half of number of cells on that colum.



a) Find an inpressive plant when $m=n=2016$;

b) Prove that if there at least a inpressive plant then $4|m$ and $4|n$.
2016 Vietnam National Olympiad
Find all $a\in\mathbb{R}$ such that there is function $f:\mathbb{R}\to\mathbb{R}$

i) $f(1)=2016$

ii) $f(x+y+f(y))=f(x)+ay\quad\forall x,y\in\mathbb{R}$
2016 Vietnam National Olympiad
Given a triangle $ABC$ inscribed by circumcircle $(O)$. The angles at $B,C$ are acute angle. Let $M$ on the arc $BC$ that doesn't contain $A$ such that $AM$ is not perpendicular to $BC$. $AM$ meets the perpendicular bisector of $BC$ at $T$. The circumcircle $(AOT)$ meets $(O)$ at $N$ ($N\ne A$).



a) Prove that $\angle{BAM}=\angle{CAN}$.



b) Let $I$ be the incenter and $G$ be the foor of the angle bisector of $\angle{BAC}$. $AI,MI,NI$ intersect $(O)$ at $D,E,F$ respectively. Let ${P}=DF\cap AM, {Q}=DE\cap AN$. The circle passes through $P$ and touches $AD$ at $I$ meets $DF$ at $H$ ($H\ne D$).The circle passes through $Q$ and touches $AD$ at $I$ meets $DE$ at $K$ ($K\ne D$). Prove that the circumcircle $(GHK)$ touches $BC$.
2016 Vietnam National Olympiad
a) Prove that if $n$ is an odd perfect number then $n$ has the following form $$ n=p^sm^2 $$where $p$ is prime has form $4k+1$, $s$ is positive integers has form $4h+1$, and $m\in\mathbb{Z}^+$, $m$ is not divisible by $p$.

b) Find all $n\in\mathbb{Z}^+$, $n>1$ such that $n-1$ and $\frac{n(n+1)}{2}$ is perfect number
2015 Vietnam National Olympiad
Given a non negative real $a$ and a sequence $(u_n)$ defined by $$ \begin{cases} u_1=3\\ u_{n+1}=\frac{u_n}{2}+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3} \end{cases} $$

a) Prove that for $a=0$, the sequence is convergent and find its limit.



b) For $a\in [0,1]$, prove that the sequence if convergent.
2015 Vietnam National Olympiad
If $a,b,c$ are nonnegative real numbers, then

$${ 3(a^2+b^2+c^2) \geq (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2 \geq (a+b+c)^2.}$$