Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 Greece National Olympiad
Let positive integers $p,q$ with $\gcd(p,q)=1$ such as $p+q^2=(n^2+1)p^2+q$. If the parameter $n$ is a positive integer, find all possible couples $(p,q)$.
2012 Greece National Olympiad
Find all the non-zero polynomials $P(x),Q(x)$ with real coefficients and the minimum degree,such that for all $x \in \mathbb{R}$:

$$ P(x^2)+Q(x)=P(x)+x^5Q(x) $$
2012 Greece National Olympiad
Let an acute-angled triangle $ABC$ with $AB<AC<BC$, inscribed in circle $c(O,R)$. The angle bisector $AD$ meets $c(O,R)$ at $K$. The circle $c_1(O_1,R_1)$(which passes from $A,D$ and has its center $O_1$ on $OA$) meets $AB$ at $E$ and $AC$ at $Z$. If $M,N$ are the midpoints of $ZC$ and $BE$ respectively, prove that:

a)the lines $ZE,DM,KC$ are concurrent at one point $T$.

b)the lines $ZE,DN,KB$ are concurrent at one point $X$.

c)$OK$ is the perpendicular bisector of $TX$.
2012 Greece National Olympiad
The following isosceles trapezoid consists of equal equilateral triangles with side length $1$.  The side $A_1E$ has length $3$ while the larger base $A_1A_n$ has length $n-1$.  Starting from the point $A_1$ we move along the segments which are oriented to the right and up(obliquely right or left). Calculate (in terms of $n$ or not) the number of all possible paths we can follow, in order to arrive at points $B,\Gamma,\Delta, E$, if $n$ is an integer greater than $3$.



[Need image]
2011 Greece National Olympiad
Solve in integers the equation

$${x^3}{y^2}\left( {2y - x} \right) = {x^2}{y^4} - 36$$
2011 Greece National Olympiad
In the Cartesian plane $Oxy$ we consider the points ${A_1}\left( {40,1} \right), {A_2}\left( {40,2} \right), \ldots , {A_{40}}\left( {40,40} \right)$ as well as the segments $O{A_1},O{A_2},\ldots,O{A_{40}}$. A point of the Cartesian plane $Oxy$ is called "good", if its coordinates are integers and it is internal of one segment $O{A_i}, i=1,2,3,\ldots,40$. Additionally, one of the segments $O{A_1},O{A_2},\ldots,O{A_{40}}$ is called "good" if  it contains a "good" point. Find the number of "good" segments and "good" points.
2011 Greece National Olympiad
Let $a,b,c$ be positive real numbers with sum $6$. Find the maximum value of

$$S = \sqrt[3]{{{a^2} + 2bc}} + \sqrt[3]{{{b^2} + 2ca}} + \sqrt[3]{{{c^2} + 2ab}}.$$
2011 Greece National Olympiad
We consider an acute angled triangle $ABC$ (with $AB<AC$) and its circumcircle $c(O,R) $(with center $O$ and semidiametre $R$).The altitude $AD$ cuts the circumcircle at the point $E$ ,while the perpedicular bisector $(m)$ of the segment $AB$,cuts $AD$ at the point $L$.$BL$ cuts $AC$ at the point $M$ and the circumcircle $c(O,R)$ at the point $N$.Finally $EN$ cuts the perpedicular bisector $(m)$ at the point $Z$.Prove that:

$$ MZ \perp BC \iff \left(CA=CB \;\; \text{or} \;\; Z\equiv O \right) $$
2010 Greece National Olympiad
Solve in the integers the diophantine equation

$$x^4-6x^2+1 = 7 \cdot 2^y.$$
2010 Greece National Olympiad
If $ x,y$ are positive real numbers with sum $ 2a$, prove that :





$ x^3y^3(x^2+y^2)^2 \leq 4a^{10}$



When does equality hold ?



Babis
2010 Greece National Olympiad
A triangle $ ABC$ is inscribed in a circle $ C(O,R)$  and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively.



Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic.





Babis
2010 Greece National Olympiad
On the plane are given $ k+n$  distinct lines , where $ k>1$ is integer and $ n$ is integer as well.Any three of these lines do not pass through the



same point . Among these lines exactly  $ k$ are parallel and all the other $ n$ lines intersect each other.All $ k+n$ lines define on the plane a partition



of triangular , polygonic or not bounded regions. Two regions are colled different, if the have not common points



or if they have common points only on their boundary.A regions is called ''good'' if it contained in a zone between two parallel lines .



If in a such given configuration the minimum number of ''good'' regionrs is $ 176$ and the maximum number of these regions is $ 221$, find $ k$ and $ n$.





Babis
2009 Greece National Olympiad
Find all positive integers $n$ such that the number $$A=\sqrt{\frac{9n-1}{n+7}}$$ is rational.
2009 Greece National Olympiad
Consider a triangle $ABC$ with circumcenter $O$ and let $A_1,B_1,C_1$ be the midpoints of the sides $BC,AC,AB,$ respectively.

Points $A_2,B_2,C_2$ are defined as $\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda  \cdot \overrightarrow{OC_1},$ where $\lambda >0.$

Prove that lines $AA_2,BB_2,CC_2$ are concurrent.
2009 Greece National Olympiad
Let $ x,y,z$ be nonnegative real numbers such that $ x + y + z = 2$. Prove that $ x^{2}y^{2} + y^{2}z^{2} + z^{2}x^{2} + xyz\leq 1$. When does the equality occur?
2009 Greece National Olympiad
Consider pairwise distinct complex numbers $z_1,z_2,z_3,z_4,z_5,z_6$ whose images $A_1,A_2,A_3,A_4,A_5,A_6$ respectively are succesive points on the circle centered at $O(0,0)$ and having radius $r>0.$

If $w$ is a root of the equation $z^2+z+1=0$ and the next equalities hold $$z_1w^2+z_3w+z_5=0 \\ z_2w^2+z_4w+z_6=0$$ prove that



a) Triangle $A_1A_3A_5$ is equilateral

b) $$|z_1-z_2|+|z_2-z_3|+|z_3-z_4|+|z_4-z_5|+z_5-z_6|+|z_6-z_1|=3|z_1-z_4|=3|z_2-z_5|=3|z_3-z_6|.$$
2008 Greece National Olympiad
A computer generates all pairs of real numbers $x, y \in (0, 1)$ for which the numbers $a = x+my$ and $b = y+mx$ are both integers, where $m$ is a given positive integer. Finding one such pair $(x, y)$ takes $5$ seconds. Find $m$ if  the computer needs $595$ seconds to find all possible ordered pairs $(x, y)$.
2008 Greece National Olympiad
Find all integers $x$ and prime numbers $p$ satisfying $x^8 + 2^{2^x+2} = p$.
2008 Greece National Olympiad
A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$, where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK -SH) = 1$, compute the area of the concave quadrilateral $ABHC$.
2008 Greece National Olympiad
If $a_1, a_2, \ldots , a_n$ are positive integers and $k = \max\{a_1, \ldots, a_n\}$, $t = \min\{a_1,\ldots, a_n\}$, prove the inequality

$$\left(\frac{a_1^2+a_2^2+\cdots+a_n^2}{a_1+a_2+\cdots+a_n}\right)^{\frac{kn}{t}} \geq a_1a_2\cdots a_n.$$

When does equality hold?
2007 Greece National Olympiad
Find all positive integers $n$ such that $4^{n}+2007$ is a perfect square.
2007 Greece National Olympiad
Let $a,b,c$ be sides of a triangle, show that

$$\frac{(c+a-b)^{4}}{a(a+b-c)}+\frac{(a+b-c)^{4}}{b(b+c-a)}+\frac{(b+c-a)^{4}}{c(c+a-b)}\geq ab+bc+ca.$$
2007 Greece National Olympiad
In a circular ring with radii $11r$ and $9r$, we put circles of radius $r$ which are tangent to the boundary circles and do not overlap. Determine the maximum number of circles that can be put this way. (You may use that $9.94<\sqrt{99}<9.95$)
2007 Greece National Olympiad
Given a $2007\times 2007$ array of numbers $1$ and $-1$, let $A_{i}$ denote the product of the entries in the $i$th row, and $B_{j}$ denote the product of the entries in the $j$th column. Show that

$$A_{1}+A_{2}+\cdots +A_{2007}+B_{1}+B_{2}+\cdots +B_{2007}\neq 0.$$
2006 Greece National Olympiad
How many 5 digit positive integers are there such that each of its digits, except for the last one, is greater than or equal to the next digit?
2006 Greece National Olympiad
Let $n$ be a positive integer. Prove that the equation

$$x+y+\frac{1}{x}+\frac{1}{y}=3n$$

does not have solutions in positive rational numbers.
2006 Greece National Olympiad
Let a triangle  $ABC$   and the cevians  $AL, BN , CM$ such that  $AL$  is the  bisector of angle  $A$. If  $\angle ALB = \angle ANM$, prove that  $\angle MNL = 90$.
2006 Greece National Olympiad
Does there exist a function $f : \mathbb{R} \rightarrow \mathbb{R}$, which satisfies both conditions :



a) $f( x + y + z) \leq 3(xy + yz + zx)$   for all real numbers $x , y , z$



and



b) there exist function $g$  and natural number $n$, such that



$g(g(x)) = x ^ {2n + 1}$    and         $f(g(x)) = (g(x)) ^2$  for every real number  $x$ ?
2005 Greece National Olympiad
Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.
2005 Greece National Olympiad
The  sequence $(a_n)$ is  defined  by  $a_1=1$ and $a_n=a_{n-1}+\frac{1}{n^3}$  for  $n>1.$

(a) Prove  that   $a_n<\frac{5}{4}$    for all  $n.$

(b) Given $\epsilon>0$, find  the smallest natural number  $n_0$  such  that  ${\mid a_{n+1}-a_n}\mid<\epsilon$  for all  $n>n_0.$
2005 Greece National Olympiad
We know that $k$ is a positive integer and the equation $$ x^3+y^3-2y(x^2-xy+y^2)=k^2(x-y) \quad (1) $$ has one solution $(x_0,y_0)$ with

$x_0,y_0\in \mathbb{Z}-\{0\}$ and $x_0\neq y_0$. Prove that



i) the equation (1) has a finite number of solutions $(x,y)$ with $x,y\in \mathbb{Z}$ and $x\neq y$;



ii) it is possible to find $11$ addition different solutions $(X,Y)$ of the equation (1) with $X,Y\in \mathbb{Z}-\{0\}$ and $X\neq Y$ where $X,Y$ are  functions of $x_0,y_0$.
2005 Greece National Olympiad
Let  $OX_1 , OX_2$  be  rays in  the interior of a  convex  angle $XOY$ such that $\angle XOX_1=\angle YOY_1< \frac{1}{3}\angle XOY$. Points  $K$  on $OX_1$  and  $L$  on  $OY_1$  are fixed so that  $OK=OL$, and  points $A$, $B$  are  vary on rays $(OX , (OY$  respectively  such that  the area  of the pentagon  $OAKLB$  remains  constant. Prove that  the  circumcircle of the  triangle $OAB$  passes  from a  fixed  point,  other than  $O$.
2004 Greece National Olympiad
Find the greatest value of $M$ $\in \mathbb{R}$ such that the following inequality is true $\forall$ $x, y, z$ $\in \mathbb{R}$

$x^4+y^4+z^4+xyz(x+y+z)\geq M(xy+yz+zx)^2$.
2004 Greece National Olympiad
If $m\geq 2$ show that there does not exist positive  integers $x_1, x_2, ..., x_m,$ such that $$x_1< x_2<...< x_m \ \ \text{and} \ \ \frac{1}{x_1^3}+\frac{1}{x_2^3}+...+\frac{1}{x_m^3}=1.$$
2004 Greece National Olympiad
Consider a circle $K(O,r)$ and a point $A$ outside $K.$ A line $\epsilon$ different from $AO$ cuts $K$ at $B$ and $C,$ where $B$ lies between $A$ and $C.$

Now the symmetric line of $\epsilon$ with respect to axis of symmetry the  line $AO$ cuts $K$ at $E$ and $D,$ where $E$ lies between $A$ and $D.$

Show that  the diagonals of the quadrilateral $BCDE$ intersect in a fixed point.
2004 Greece National Olympiad
Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$

If no element of $M$ is equal to the sum of any two elements of $M,$

find the least value that the greatest element of $M$ can take.
2003 Greece National Olympiad
If $a, b, c, d$ are positive numbers satisfying $a^3 + b^3 +3ab = c + d = 1,$ prove that $$\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.$$
2003 Greece National Olympiad
Find all real solutions of the system $$\begin{cases}x^2 + y^2 - z(x + y) = 2, \\ y^2 + z^2 - x(y + z) = 4, \\ z^2 + x^2 - y(z + x) = 8.\end{cases}$$
2003 Greece National Olympiad
Given are a circle $\mathcal{C}$ with center $K$ and radius $r,$ point  $A$ on the circle and point $R$ in its exterior. Consider a variable line $e$ through $R$ that intersects the circle at two points $B$ and $C.$  Let $H$ be the orthocenter of triangle $ABC.$



Show that there is a unique point $T$ in the plane of circle $\mathcal{C}$ such that the sum $HA^2 + HT^2$ remains constant (as $e$ varies.)
2003 Greece National Olympiad
On the set $\Sigma$ of points of the plane $\Pi$ we deﬁne the operation $*$ which maps each pair $(X, Y )$ of points in $\Sigma$ to the point $Z = X * Y$ that is symmetric to $X$ with respect to $Y .$ Consider a square $ABCD$ in $\Pi$. Is it possible, using the points $A, B, C$ and applying the operation $*$ ﬁnitely many times, to construct the point $D?$
2002 Greece National Olympiad
The real numbers  $a,b,c$  with  $bc\neq0$  satisfy $\frac{1-c^2}{bc}\geq0.$ Prove  that  $10(a^2+b^2+c^2-bc^3)\geq2ab+5ac.$
2002 Greece National Olympiad
A  student of the  National  Technical  University  was  reading  advanced mathematics  last summer for  37  days  according  to the following rules :

(a) He was  reading  at least  one hour  every  day.

(b) He was reading an integer  number of  hours, but not more  than 12, each day.

(c) He  had  to read at most  60  hours in total.



Prove  that there  were some  successive  days  during  which  the student was  reading  exactly 13  hours in  total.
2002 Greece National Olympiad
In a  triangle $ABC$  we have  $\angle C>10^0$ and  $\angle B=\angle C+10^0.$We  consider  point  $E$  on side  $AB$  such that  $\angle ACE=10^0,$ and point  $D$ on side $AC$  such that  $\angle DBA=15^0.$ Let  $Z\neq A$ be a point  of interection of  the  circumcircles of  the triangles  $ABD$ and $AEC.$Prove that  $\angle ZBA>\angle ZCA.$
2002 Greece National Olympiad
(a) Positive  integers  $p,q,r,a$  satisfy  $pq=ra^2$, where  $r$  is prime  and $p,q$ are  relatively  prime. Prove  that one  of the  numbers  $p,q$ is a  perfect  square.

(b) Examine if there  exists  a  prime  $p$  such that  $p(2^{p+1}-1)$ is a  perfect square.
2001 Greece National Olympiad
A triangle $ABC$ is inscribed in a circle of radius $R.$ Let $BD$ and $CE$ be the bisectors of the angles $B$ and $C$ respectively and let the line $DE$ meet the arc $AB$ not containing $C$ at point $K.$ Let $A_1, B_1, C_1$ be the feet of perpendiculars from $K$ to $BC, AC, AB,$ and $x, y$ be the distances from $D$ and $E$ to $BC,$ respectively.



(a) Express the lengths of $KA_1, KB_1, KC_1$ in terms of $x, y$ and the ratio $l = KD/ED.$



(b) Prove that $\frac{1}{KB}=\frac{1}{KA}+\frac{1}{KC}.$
2001 Greece National Olympiad
Prove that there are no positive integers $a,b$ such that $(15a +b)(a +15b)$ is a power of $3.$
2001 Greece National Olympiad
A function $f : \Bbb{N}_0 \to \Bbb{R}$ satisﬁes $f(1) = 3$ and $$f(m + n) + f(m - n) - m + n - 1 =\frac{f(2m) + f(2n)}{2},$$for any non-negative integers $m$ and $n$ with $m \geq n.$ Find all such functions $f$.
2001 Greece National Olympiad
The numbers $1$ to $500$ are written on a board. Two pupils $A$ and $B$ play the following game: A player in turn deletes one of the numbers from the board. The game is over when only two numbers remain. Player $B$ wins if the sum of the two remaining numbers is divisible by $3,$ otherwise $A$ wins. If $A$ plays ﬁrst, show that $B$ has a winning strategy.
2000 Greece National Olympiad
Consider a rectangle $ABCD$ with $AB = a$ and $AD = b.$ Let $l$ be a line through $O,$ the center of the rectangle, that cuts $AD$ in  $E$ such that $AE/ED = 1/2$. Let $M$ be any point on $l,$ interior to the rectangle.

Find the necessary and suﬃcient condition on $a$ and $b$ that the four distances from M to lines $AD, AB, DC, BC$ in this order form an arithmetic progression.
2000 Greece National Olympiad
Find all prime numbers $p$ such that $1 +p+p^2 +p^3 +p^4$ is a perfect square.
2000 Greece National Olympiad
Find the maximum value of $k$ such that $$\frac{xy}{\sqrt{(x^2 + y^2)(3x^2 + y^2)}}\leq \frac{1}{k}$$

holds for all positive numbers $x$ and $y.$
2000 Greece National Olympiad
The subsets $A_1,A_2,\ldots ,A_{2000}$ of a finite set $M$ satisfy $|A_i|>\frac{2}{3}|M|$ for each $i=1,2,\ldots ,2000$. Prove that there exists $m\in M$ which belongs to at least $1334$ of the subsets $A_i$.
1999 Greece National Olympiad
Let $f(x)=ax^2+bx+c$, where $a,b,c$ are nonnegative real numbers, not all equal to zero. Prove that $f(xy)^2\le f(x^2)f(y^2)$ for all real numbers $x,y$.
1999 Greece National Olympiad
A right triangle has integer side lengths, and the sum of its area and the length of one of its legs equals $75$. Find the side lengths of the triangle.
1999 Greece National Olympiad
In an acute-angled triangle $ABC$, $AD,BE$ and $CF$ are the altitudes and $H$ the orthocentre. Lines $EF$ and $BC$ meet at $N$. The line passing through $D$ and parallel to $FE$ meets lines $AB$ and $AC$ at $K$ and $L$, respectively. Prove that the circumcircle of the triangle $NKL$ bisects the side $BC$.
1999 Greece National Olympiad
On a circle are given $n\ge 3$ points. At most, how many parts can the segments with the endpoints at these $n$ points divide the interior of the circle into?
1998 Greece National Olympiad
Prove that for any integer $n>3$ there exist infinitely many non-constant arithmetic progressions of length $n-1$ whose terms are positive integers whose product is a perfect $n$-th power.
1998 Greece National Olympiad
For a regular $n$-gon, let $M$ be the set of the lengths of the segments joining its vertices. Show that the sum of the squares of the elements of $M$ is greater than twice the area of the polygon.
1998 Greece National Olympiad
Prove that for any non-zero real numbers $a, b, c,$

$$\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac 35.$$
1998 Greece National Olympiad
Let a function $g:\mathbb{N}_0\to\mathbb{N}_0$ satisfy $g(0)=0$ and $g(n)=n-g(g(n-1))$ for all $n\ge 1$. Prove that:



a) $g(k)\ge g(k-1)$ for any positive integer $k$.

b) There is no $k$ such that $g(k-1)=g(k)=g(k+1)$.
1997 Greece National Olympiad
Let $P$ be a point inside or on the boundary of a square $ABCD$. Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$.
1997 Greece National Olympiad
Let a function $f : \Bbb{R}^+ \to \Bbb{R}$ satisfy:

(i) $f$ is strictly increasing,

(ii) $f(x) > -1/x$ for all $x > 0$,

(iii)$ f(x)f (f(x) + 1/x) = 1$ for all $x > 0$.

Determine $f(1)$.
1997 Greece National Olympiad
Find all integer solutions to $$\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997}.$$
1997 Greece National Olympiad
A polynomial $P$ with integer coefficients has at least $13$ distinct integer roots. Prove that if an integer $n$ is not a root of $P$, then $|P(n)| \geq 7 \cdot 6!^2$, and give an example for equality.
1996 Greece National Olympiad
Let $a_n$ be a sequence of positive numbers such that:

i) $\dfrac{a_{n+2}}{a_n}=\dfrac{1}{4}$, for every $n\in\mathbb{N}^{\star}$

ii) $\dfrac{a_{k+1}}{a_k}+\dfrac{a_{n+1}}{a_n}=1$, for every $ k,n\in\mathbb{N}^{\star}$ with $|k-n|\neq 1$.

(a) Prove that $(a_n)$ is a geometric progression.

(n) Prove that exists  $t>0$, such that $\sqrt{a_{n+1}}\leq \dfrac{1}{2}a_n+t$
1996 Greece National Olympiad
Let $ ABC$ be an acute triangle, $ AD,BE,CZ$ its altitudes and $ H$ its orthocenter. Let $ AI,A \Theta$ be the internal and external bisectors of angle $ A$. Let $ M,N$ be the midpoints of $ BC,AH$, respectively. Prove that:

(a) $MN$ is perpendicular to $EZ$

(b) if $ MN$ cuts the segments $ AI,A \Theta$ at the points $ K,L$, then $ KL=AH$
1996 Greece National Olympiad
Prove that among $81$ natural numbers whose prime divisors are in the set $\{2, 3, 5\}$ there exist four numbers whose product is the fourth power of an integer.
1996 Greece National Olympiad
Find the number of functions $f : \{1, 2, . . . , n\} \to \{1995, 1996\}$ such that $f(1) + f(2) + ... + f(1996)$ is odd.
1995 Greece National Olympiad
Find all positive integers $n$ such that $-5^4 + 5^5 + 5^n$ is a perfect square. Do the same for $2^4 + 2^7 + 2^n.$
1995 Greece National Olympiad
Let $ABC$ be a triangle with $AB = AC$ and let $D$ be a point on $BC$ such that the incircle of $ABD$ and the excircle of $ADC$ corresponding to $A$ have the same radius. Prove that this radius is equal to one quarter of the altitude from $B$ of triangle $ABC$.
1995 Greece National Olympiad
If the equation $ ax^2+(c-b)x+(e-d)=0$ has real roots greater than $1$, prove that the equation $ ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.
1995 Greece National Olympiad
Given are the lines $l_1,l_2,\ldots ,l_k$ in the plane, no two of which are parallel and no three of which are concurrent. For which $k$ can one label the intersection points of these lines by $1, 2,\ldots , k-1$ so that in each of the given lines all the labels appear exactly once?
2013 Hong kong National Olympiad
Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that

$$\sqrt[4]{\frac{\sqrt{3}}{a}+6\sqrt{3}b}+\sqrt[4]{\frac{\sqrt{3}}{b}+6\sqrt{3}c}+\sqrt[4]{\frac{\sqrt{3}}{c}+6\sqrt{3}a}\le\frac{1}{abc}$$

When does inequality hold?
2013 Hong kong National Olympiad
For any positive integer $a$, define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$. Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$.
2013 Hong kong National Olympiad
Let $ABC$ be a triangle with $CA>BC>AB$. Let $O$ and $H$ be the circumcentre and orthocentre of triangle $ABC$ respectively. Denote by $D$ and $E$ the midpoints of the arcs $AB$ and $AC$ of the circumcircle of triangle $ABC$ not containing the opposite vertices. Let $D'$ be the reflection of $D$ about $AB$ and $E'$ the reflection of $E$ about $AC$. Prove that $O,H,D',E'$ are concylic if and only if $A,D',E'$ are collinear.
2013 Hong kong National Olympiad
In a chess tournament there are $n>2$ players. Every two players play against each other exactly once. It is known that exactly $n$ games end as a tie. For any set $S$ of players, including $A$ and $B$, we say that $A$ admires $B$ in that set if

i) $A$ does not beat $B$; or

ii) there exists a sequence of players $C_1,C_2,\ldots,C_k$ in $S$, such that $A$ does not beat $C_1$, $C_k$ does not beat $B$, and $C_i$ does not beat $C_{i+1}$ for $1\le i\le k-1$.

A set of four players is said to be harmonic if each of the four players admires everyone else in the set. Find, in terms of $n$, the largest possible number of harmonic sets.
2010 Hong kong National Olympiad
Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral.
2010 Hong kong National Olympiad
Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition:  for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds $$\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2$$
2010 Hong kong National Olympiad
Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that

$$\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}$$

where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.
2009 Hong kong National Olympiad
let ${a_{n}}$ be a sequence of integers,$a_{1}$ is odd,and for any positive integer $n$,we have

$n(a_{n+1}-a_{n}+3)=a_{n+1}+a_{n}+3$,in addition,we have $2010$ divides $a_{2009}$

find the smallest $n\ge\ 2$,so that $2010$ divides $a_{n}$
2009 Hong kong National Olympiad
there are $n$ points on the plane,any two vertex are connected by an edge of red,yellow or green,and any triangle with vertex in the graph contains exactly $2$ colours.prove that $n<13$
2009 Hong kong National Olympiad
$ABC$ is a right triangle with $\angle C=90$,$CD$ is perpendicular to $AB$,and $D$ is the foot,$\omega$ is the circumcircle of triangle $BCD$,$\omega_{1}$ is a circle inside triangle $ACD$,tangent to $AD$ and $AC$ at $M$ and $N$ respectively,and $\omega_{1}$ is also tangent to $\omega$.prove that:

(1)$BD*CN+BC*DM=CD*BM$

(2)$BM=BC$
2009 Hong kong National Olympiad
find all pairs of non-negative integer pairs $(m,n)$,satisfies

$107^{56}(m^{2}-1)+2m+3=\binom{113^{114}}{n}$
2008 Hong kong National Olympiad
Let $ f(x) = c_m x^m + c_{m-1} x^{m-1} +...+ c_1 x + c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 = 0$ and $ a_{n+1} = f(a_n)$ for all positive integers $ n$.



(a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j+1} - a_j$ is a multiple of $ a_{i+1} - a_i$.



(b) Show that $ a_{2008} \neq 0$