Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 South East Mathematical Olympiad
Can you make $2015$ positive integers $1,2, \ldots , 2015$ to be a certain permutation which can be ordered in the circle such that the sum of any two adjacent numbers is a multiple of $4$ or a multiple of $7$?
2015 South East Mathematical Olympiad
For any positive integer $n$, we have the set $P_n = \{ n^k \mid k=0,1,2, \ldots \}$. For positive integers $a,b,c$, we define the group of $(a,b,c)$ as lucky if there is a positive integer $m$ such that $a-1$, $ab-12$, $abc-2015$ (the three numbers need not be different from each other) belong to the set $P_m$. Find the number of lucky groups.
2015 South East Mathematical Olympiad
Suppose that $a,b$ are real numbers, function $f(x) = ax+b$ satisfies $\mid f(x) \mid \leq 1$ for any $x \in [0,1]$. Find the range of values of $S= (a+1)(b+1).$
2015 South East Mathematical Olympiad
In $\triangle ABC$, we have three edges with lengths $BC=a, \, CA=b \, AB=c$, and $c<b<a<2c$. $P$ and $Q$ are two points of the edges of $\triangle ABC$, and the straight line $PQ$ divides $\triangle ABC$ into two parts with the same area. Find the minimum value of the length of the line segment $PQ$.
2015 South East Mathematical Olympiad
In $\triangle ABC$, we have $AB>AC>BC$. $D,E,F$ are the tangent points of the inscribed circle of $\triangle ABC$ with the line segments $AB,BC,AC$ respectively. The points $L,M,N$ are the midpoints of the line segments $DE,EF,FD$. The straight line $NL$ intersects with ray $AB$ at $P$, straight line $LM$ intersects ray $BC$ at $Q$ and the straight line $NM$ intersects ray $AC$ at $R$. Prove that $PA \cdot QB \cdot RC = PD \cdot QE \cdot RF$.
2015 South East Mathematical Olympiad
For any integers $m,n$, we have the set $A(m,n) = \{ x^2+mx+n \mid x \in \mathbb{Z} \}$, where $\mathbb{Z}$ is the integer set. Does there exist three distinct elements $a,b,c$ which belong to $A(m,n)$ and satisfy the equality $a=bc$?
2015 South East Mathematical Olympiad
Given a sequence $\{ a_n\}_{n\in \mathbb{Z}^+}$ defined by $a_1=1$ and $a_{2k}=a_{2k-1}+a_k,a_{2k+1}=a_{2k}$ for all positive integer $k$.

Prove that, for any positive integer $n$, $a_{2^n}>2^{\frac{n^2}{4}}$.
2015 South East Mathematical Olympiad
Given $8$ pairwise distinct positive integers $a_1,a_2,…,a_8$ such that the greatest common divisor of any three of them is equal to $1$. Show that there exists positive integer $n\geq 8$ and $n$ pairwise distinct positive integers $m_1,m_2,…,m_n$ with the greatest common divisor of all $n$ numbers equal to $1$ such that for any positive integers $1\leq p<q<r\leq n$, there exists positive integers $1\leq i<j\leq 8$ that $a_ia_j\mid m_p+m_q+m_r$.
2015 South East Mathematical Olympiad
Given two points $E$ and $F$ lie on segment $AB$ and $AD$, respectively. Let the segments $BF$ and $DE$ intersects at point $C$. If it’s known that $AE+EC=AF+FC$, show that $AB+BC=AD+DC$.
2015 South East Mathematical Olympiad
Given a positive integer $n\geq 2$. Let $A=\{ (a,b)\mid a,b\in \{ 1,2,…,n\} \}$ be the set of points in Cartesian coordinate plane. How many ways to colour points in $A$, each by one of three fixed colour, such that, for any $a,b\in \{ 1,2,…,n-1\}$, if $(a,b)$ and $(a+1,b)$ have same colour, then $(a,b+1)$ and $(a+1,b+1)$ also have same colour.
2015 South East Mathematical Olympiad
Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$, where $i$ is the imaginary unit.
2014 South East Mathematical Olympiad
Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$
2014 South East Mathematical Olympiad
Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$
2014 South East Mathematical Olympiad
In an obtuse triangle $ABC$ $(AB>AC)$,$O$ is the circumcentre and $D,E,F$ are the midpoints of $BC,CA,AB$ respectively.Median $AD$ intersects $OF$ and $OE$ at $M$ and $N$ respectively.$BM$ meets $CN$ at point $P$.Prove that $OP\perp AP$
2014 South East Mathematical Olympiad
Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$. Prove that$$x_1+x_2+\cdots+x_n\le \frac{5}{3}.$$
2014 South East Mathematical Olympiad
Let $\triangle ABC $ and $\triangle A'B'C'$ are acute triangles.Prove that$$Max\{cotA'(cotB+cotC),cotB'(cotC+cotA),cotC'(cotA+cotB)\}\ge \frac{2}{3}.$$
2014 South East Mathematical Olympiad
Let $a,b$ and $c$ be integers and $r$ a real number such that $ar^2+br+c=0$ with $ac\not =0$.Prove that $\sqrt{r^2+c^2}$ is an irrational number
2014 South East Mathematical Olympiad
Let $\omega_{1}$ be a circle with centre $O$. $P$ is a point on $\omega_{1}$. $\omega_{2}$ is a circle with centre $P$, with radius smaller than $\omega_{1}$. $\omega_{1}$ meets $\omega_{2}$ at points $T$ and $Q$. Let $TR$ be a diameter of $\omega_{2}$. Draw another two circles with $RQ$ as the radius, $R$ and $P$ as the centres. These two circles meet at point $M$, with $M$ and $Q$ lie on the same side of $PR$. A circle with centre $M$ and radius $MR$ intersects $\omega_{2}$ at $R$ and $N$. Prove that a circle with centre $T$ and radius $TN$ passes through $O$.
2014 South East Mathematical Olympiad
Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions:

$(1)$Square bar $AB$ is bisected by square bar $CD$

$(2)$At least one square of $AB$ lay on both sides of $CD$

$(3)$At least one square of $CD$ lay on both sides of $AB$



There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet
2014 South East Mathematical Olympiad
Let $p$ be a primes ,$x,y,z $ be positive integers such that $x<y<z<p$ and $\{\frac{x^3}{p}\}=\{\frac{y^3}{p}\}=\{\frac{z^3}{p}\}$.

Prove that $(x+y+z)|(x^5+y^5+z^5).$
2014 South East Mathematical Olympiad
Let $x_1,x_2,\cdots,x_n$ be positive  real numbers such that $x_1+x_2+\cdots+x_n=1$ $(n\ge 2)$. Prove that$$\sum_{i=1}^n\frac{x_i}{x_{i+1}-x^3_{i+1}}\ge \frac{n^3}{n^2-1}.$$here $x_{n+1}=x_1.$
2014 South East Mathematical Olympiad
Show that there are infinitely  many triples  of positive integers $(a_i,b_i,c_i)$, $i=1,2,3,\ldots$, satisfying the equation $a^2+b^2=c^4$, such that $c_n$ and $c_{n+1}$ are coprime for any positive integer $n$.
2013 South East Mathematical Olympiad
Let $a,b$ be real numbers such that the equation $x^3-ax^2+bx-a=0$ has three  positive real roots . Find the minimum of $\frac{2a^3-3ab+3a}{b+1}$.
2013 South East Mathematical Olympiad
$\triangle ABC$, $AB>AC$. the incircle $I$ of $\triangle ABC$ meet $BC$ at point $D$, $AD$ meet $I$ again at $E$. $EP$ is a tangent of $I$, and $EP$ meet the extension line of $BC$ at $P$. $CF\parallel PE$, $CF\cap AD=F$. the line $BF$ meet $I$ at $M,N$, point $M$ is on the line segment $BF$, the line segment $PM$ meet $I$ again at $Q$. Show that $\angle ENP=\angle ENQ$
2013 South East Mathematical Olympiad
A sequence $\{a_n\}$ , $a_1=1,a_2=2,a_{n+1}=\dfrac{a_n^2+(-1)^n}{a_{n-1}}$. Show that $a_m^2+a_{m+1}^2\in\{a_n\},\forall m\in\Bbb N$
2013 South East Mathematical Olympiad
There are $12$ acrobats who are assigned a distinct number ($1, 2, \cdots , 12$) respectively. Half of them stand around forming a circle (called circle A); the rest form another circle (called circle B) by standing on the shoulders of every two adjacent acrobats in circle A respectively. Then circle A and circle B make up a formation. We call a formation a “tower” if the number of any acrobat in circle B is equal to the sum of the numbers of the two acrobats whom he stands on. How many heterogeneous towers are there?

(Note: two towers are homogeneous if either they are symmetrical or one may become the other one by rotation. We present an example of $8$ acrobats (see attachment). Numbers inside the circle represent the circle A; numbers outside the circle represent the circle B. All these three formations are “towers”, however they are homogeneous towers.)
2013 South East Mathematical Olympiad
$f(x)=\sum\limits_{i=1}^{2013}\left[\dfrac{x}{i!}\right]$. A integer $n$ is called good if $f(x)=n$ has real root. How many good numbers are  in $\{1,3,5,\dotsc,2013\}$?
2013 South East Mathematical Olympiad
$n>1$ is an integer. The first $n$ primes are $p_1=2,p_2=3,\dotsc, p_n$. Set $A=p_1^{p_1}p_2^{p_2}...p_n^{p_n}$.

Find all positive integers $x$, such that $\dfrac Ax$ is even, and $\dfrac Ax$ has exactly $x$  divisors
2013 South East Mathematical Olympiad
Given a $3\times 3$ grid, we call the remainder of the grid an “angle” when a $2\times 2$ grid is cut out from the grid. Now we place some angles on a $10\times 10$ grid such that the borders of those angles must lie on the grid lines or its borders, moreover there is no overlap among the angles. Determine the maximal value of $k$, such that no matter how we place $k$ angles on the grid, we can always place another angle on the grid.
2013 South East Mathematical Olympiad
$n\geq 3$ is a integer. $\alpha,\beta,\gamma \in (0,1)$.  For every $a_k,b_k,c_k\geq0(k=1,2,\dotsc,n)$ with $\sum\limits_{k=1}^n(k+\alpha)a_k\leq \alpha, \sum\limits_{k=1}^n(k+\beta)b_k\leq \beta, \sum\limits_{k=1}^n(k+\gamma)c_k\leq \gamma$,  we always have  $\sum\limits_{k=1}^n(k+\lambda)a_kb_kc_k\leq \lambda$.

Find the minimum  of $\lambda$
2012 South East Mathematical Olympiad
Find a triple $(l, m, n)$ of positive integers $(1<l<m<n)$, such that $\sum_{k=1}^{l}k, \sum_{k=l+1}^{m}k, \sum_{k=m+1}^{n}k$ form a geometric sequence in order.
2012 South East Mathematical Olympiad
The incircle $I$ of $\triangle ABC$ is tangent to sides $AB,BC,CA$ at $D,E,F$ respectively. Line $EF$ intersects lines $AI,BI,DI$ at $M,N,K$ respectively. Prove that $DM\cdot KE=DN\cdot KF$.
2012 South East Mathematical Olympiad
For composite number $n$, let $f(n)$ denote the sum of the least three divisors of $n$, and $g(n)$ the sum of the greatest two divisors of $n$. Find all composite numbers $n$, such that $g(n)=(f(n))^m$ ($m\in N^*$).
2012 South East Mathematical Olympiad
Let $a, b, c, d$ be real numbers satisfying inequality $a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1$ holds for arbitrary real number $x$. Find the maximal value of $a+b-c+d$ and determine the values of $a,b,c,d$ when that maximum is attained.
2012 South East Mathematical Olympiad
A nonnegative integer $m$ is called a “six-composited number” if $m$ and the sum of its digits are both multiples of $6$. How many “six-composited numbers” that are less than $2012$ are there?
2012 South East Mathematical Olympiad
Find the least natural number $n$, such that the following inequality holds:$\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}$.
2012 South East Mathematical Olympiad
In $\triangle ABC$, point $D$ lies on side $AC$ such that $\angle ABD=\angle C$. Point $E$ lies on side $AB$ such that $BE=DE$. $M$ is the midpoint of segment $CD$. Point $H$ is the foot of the perpendicular from $A$ to $DE$. Given $AH=2-\sqrt{3}$ and $AB=1$, find the size of $\angle AME$.
2012 South East Mathematical Olympiad
Let positive integers $m,n$ satisfy $n=2^m-1$. $P_n =\{1,2,\cdots ,n\}$ is a set that contains $n$ points on an axis. A grasshopper on the axis can leap from one point to another adjacent point. Find the maximal value of $m$ satisfying following conditions:

(a) $x, y$ are two arbitrary points in $P_n$;

(b) starting at point $x$, the grasshopper leaps $2012$ times and finishes at point $y$; (the grasshopper is allowed to travel $x$ and $y$ more than once)

(c)  there are even number ways for the grasshopper to do (b).
2011 South East Mathematical Olympiad
If $\min \left \{ \frac{ax^2+b}{\sqrt{x^2+1}} \mid x \in \mathbb{R}\right \} = 3$, then (1) Find the range of $b$; (2) for every given $b$, find $a$.
2011 South East Mathematical Olympiad
If positive integers, $a,b,c$ are pair-wise co-prime, and, $$\ a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3) $$ find $a,b,$ and $c$
2011 South East Mathematical Olympiad
Find all positive integer $n$ , such that for all 35-element-subsets of $M=(1,2,3,...,50)$ ,there exists at least two different elements $a,b$ , satisfing : $a-b=n$ or $a+b=n$.
2011 South East Mathematical Olympiad
Let $O$ be the circumcenter of triangle $ABC$ , a line passes through $O$ intersects sides $AB,AC$ at points $M,N$ , $E$ is the midpoint of $MC$ , $F$ is the midpoint of $NB$ , prove that : $\angle FOE= \angle BAC$
2011 South East Mathematical Olympiad
In triangle $ABC$ , $AA_0,BB_0,CC_0$ are the angle bisectors , $A_0,B_0,C_0$are on sides $BC,CA,AB,$ draw $A_0A_1//BB_0,A_0A_2//CC_0$ ,$A_1$ lies on $AC$ ,$A_2$ lies on $AB$ , $A_1A_2$ intersects $BC$ at $A_3$.$B_3$ ,$C_3$ are constructed similarly.Prove that : $A_3,B_3,C_3$ are collinear.
2011 South East Mathematical Olympiad
Let $P_i$ $i=1,2,......n$ be $n$ points on the plane , $M$ is a point on segment $AB$ in the same plane , prove : $\sum_{i=1}^{n} |P_iM| \le \max( \sum_{i=1}^{n} |P_iA| ,  \sum_{i=1}^{n} |P_iB| )$. (Here $|AB|$ means the length of segment $AB$) .
2011 South East Mathematical Olympiad
The sequence $(a_n)_{n>=1}$ satisfies that : $a_1=a_2=1$ $a_n=7a_{n-1}-a_{n-2}$ ($n>=3$) , prove that : for all positive integer n , number $a_n+2+a_{n+1}$ is a perfect square .
2011 South East Mathematical Olympiad
12 points are located on a clock with the sme distance , numbers $1,2,3 , ... 12$ are marked on each point in clockwise order . Use 4 kinds of colors (red,yellow,blue,green) to colour the the points , each kind of color has 3 points . N ow , use these 12 points as the vertex of convex quadrilateral to construct $n$ convex quadrilaterals . They satisfies the following conditions:

(1). the colours of vertex of every convex quadrilateral are different from each other .

(2). for every 3 quadrilaterals among them  , there exists a colour such that : the numbers on the 3 points  painted into this colour are different from each other .

Find the maximum $n$ .
2010 South East Mathematical Olympiad
Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.
2010 South East Mathematical Olympiad
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.
2010 South East Mathematical Olympiad
The incircle of triangle $ABC$ touches $BC$ at $D$ and $AB$ at $F$, intersects the line $AD$ again at $H$ and the line $CF$ again at $K$. Prove that $\frac{FD\times HK}{FH\times DK}=3$
2010 South East Mathematical Olympiad
Let $a$ and $b$ be positive integers such that  $1\leq a<b\leq 100$. If there exists a positive integer $k$ such that $ab|a^k+b^k$, we say that the pair $(a, b)$ is good. Determine the number of good pairs.
2010 South East Mathematical Olympiad
$ABC$ is a triangle with a right angle at $C$. $M_1$ and $M_2$ are two arbitrary points inside $ABC$, and $M$ is the midpoint of $M_1M_2$. The extensions of $BM_1,BM$ and $BM_2$ intersect $AC$ at $N_1,N$ and $N_2$ respectively.

Prove that $\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}$
2010 South East Mathematical Olympiad
Let $\mathbb{N}^*$ be the set of positive integers. Define $a_1=2$, and for $n=1, 2, \ldots,$$$

a_{n+1}=\min\{\lambda|\frac{1}{a_1}+\frac{1}{a_2}+\cdots\frac{1}{a_n}+\frac{1}{\lambda}<1,\lambda\in \mathbb{N}^*\}$$

Prove that $a_{n+1}=a_n^2-a_n+1$ for $n=1,2,\ldots$.
2010 South East Mathematical Olympiad
Let $n$ be a positive integer. The real numbers $a_1,a_2,\cdots,a_n$ and $r_1,r_2,\cdots,r_n$ are such that $a_1\leq a_2\leq \cdots \leq a_n$ and $0\leq r_1\leq r_2\leq \cdots \leq r_n$.

Prove that $\sum_{i=1}^n\sum_{j=1}^n a_i a_j \min (r_i,r_j)\geq 0$
2010 South East Mathematical Olympiad
$A_1,A_2,\cdots,A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side.
2009 South East Mathematical Olympiad
Find all pairs ($x,y$) of integers such that $x^2-2xy+126y^2=2009$.
2009 South East Mathematical Olympiad
In the convex pentagon $ABCDE$ we know that $AB=DE, BC=EA$ but $AB \neq EA$.

$B,C,D,E$ are concyclic .

Prove that $A,B,C,D$ are concyclic if and only if  $AC=AD.$
2009 South East Mathematical Olympiad
Let $x,y,z $ be positive reals such that $\sqrt{a}=x(y-z)^2$, $\sqrt{b}=y(z-x)^2$ and $\sqrt{c}=z(x-y)^2$. Prove that

$$a^2+b^2+c^2 \geq 2(ab+bc+ca)$$
2009 South East Mathematical Olympiad
Given 12 red points on a circle , find the mininum value of $n$ such that there exists $n$ triangles whose vertex are the red points .

Satisfies: every chord whose points are the red points is the edge of one of the $n$ triangles .
2009 South East Mathematical Olympiad
Let $X=(x_1,x_2,......,x_9)$ be a permutation of the set $\{1,2,\ldots,9\}$ and let $A$ be the set of all such $X$ .

For any $X \in A$, denote $f(X)=x_1+2x_2+\cdots+9x_9$ and $ M=\{f(X)|X \in A \}$. Find $|M|$. ($|S|$ denotes number of members of the set $S$.)
2009 South East Mathematical Olympiad
Let $\odot O$ , $\odot I$ be the circumcircle and inscribed circles of triangle$ABC$ . Prove that : From every point $D$ on $\odot 

O$ ,we can construct a triangle $DEF$ such that $ABC$ and $DEF$ have the same circumcircle and inscribed circles
2009 South East Mathematical Olympiad
Let $x,y,z\geq0$ be real numbers such that $x+y+z=1$ Define $f(x,y,z)$ in this way :

$$f(x,y,z)=\frac{x(2y-z)}{1+x+3y}+\frac{y(2z-x)}{1+y+3z}+\frac{z(2x-y)}{1+z+3x}$$

Find the minimum value and maximum value of $f(x,y,z)$  .
2009 South East Mathematical Olympiad
In an ${8}$×${8}$ squares chart , we dig out $n$ squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart .

Then find the mininum value of $n$ .
2007 South East Mathematical Olympiad
Determine the number of real number $a$, such that for every $a$, equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:

(a) $x_0$ is an even integer;

(b) $|x_0|<1000$.
2007 South East Mathematical Olympiad
$AB$ is the diameter of semicircle $O$. $C$,$D$ are two arbitrary points on semicircle $O$. Point $P$ lies on line $CD$ such that line $PB$ is tangent to semicircle $O$ at $B$. Line $PO$ intersects line $CA$, $AD$ at point $E$, $F$ respectively. Prove that $OE$=$OF$.
2007 South East Mathematical Olympiad
Let $a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}$, determine the value of $S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]$, where $n\ge 2$ . ($[x]$ denotes the greatest integer not exceeding x)
2007 South East Mathematical Olympiad
A sequence of positive integers with $n$ terms satisfies $\sum_{i=1}^{n} a_i=2007$. Find the least positive integer $n$ such that there exist some consecutive terms in the sequence with their sum equal to $30$.
2007 South East Mathematical Olympiad
Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$.In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$. Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$.
2007 South East Mathematical Olympiad
In right-angle triangle $ABC$, $\angle C=90$°, Point $D$ is the midpoint of side $AB$. Points $M$ and $C$ lie on the same side of $AB$ such that $MB\bot AB$, line $MD$ intersects side $AC$ at $N$, line $MC$ intersects side $AB$ at $E$. Show that $\angle DBN=\angle BCE$.
2007 South East Mathematical Olympiad
Find all triples $(a,b,c)$ satisfying the following conditions:

(i)  $a,b,c$ are prime numbers, where $a<b<c<100$.

(ii) $a+1,b+1,c+1$ form a geometric sequence.
2007 South East Mathematical Olympiad
Let $a$,$b$,$c$ be positive real numbers satisfying $abc=1$. Prove that inequality $\dfrac{a^k}{a+b}+ \dfrac{b^k}{b+c}+\dfrac{c^k}{c+a}\ge \dfrac{3}{2}$ holds for all integer $k$ ($k \ge 2$).
2006 South East Mathematical Olympiad
Suppose $a>b>0$, $f(x)=\dfrac{2(a+b)x+2ab}{4x+a+b}$. Show that there exists an unique positive number $x$, such that $f(x)=\left(\dfrac{a^{\frac{1}{3}}+b^{\frac{1}{3}}}{2} \right)^3$.
2006 South East Mathematical Olympiad
In $\triangle ABC$, $\angle ABC=90^{\circ}$. Points $D,G$ lie on side $AC$. Points $E, F$ lie on segment $BD$, such that $AE \perp BD $ and $GF \perp BD$. Show that if $BE=EF$, then $\angle ABG=\angle DFC$.
2006 South East Mathematical Olympiad
There is a standard deck of $52$ cards without jokers. The deck consists of four suits(diamond, club, heart, spade) which include thirteen cards in each. For each suit, all thirteen cards are ranked from “$2$” to “$A$” (i.e. $2, 3,\ldots , Q, K, A$). A pair of cards is called a “straight flush” if these two cards belong to the same suit and their ranks are adjacent. Additionally, "$A$" and "$2$" are considered to be adjacent (i.e. "A" is also considered as "$1$"). For example, spade $A$ and spade $2$ form a “straight flush”; diamond $10$ and diamond $Q$ are not a “straight flush” pair. Determine how many ways of picking thirteen cards out of the deck such that all ranks are included but no “straight flush” exists in them.
2006 South East Mathematical Olympiad
Given any positive integer $n$, let $a_n$ be the real root of equation $x^3+\dfrac{x}{n}=1$.  Prove that

(1) $a_{n+1}>a_n$;

(2) $\sum_{i=1}^{n}\frac{1}{(i+1)^2a_i} <a_n$.
2006 South East Mathematical Olympiad
In $\triangle ABC$, $\angle A=60^\circ$. $\odot I$ is the incircle of $\triangle ABC$. $\odot I$ is tangent to sides $AB$, $AC$ at $D$, $E$, respectively. Line $DE$ intersects line $BI$ and $CI$ at $F$, $G$ respectively. Prove that $FG=\frac{BC}{2}$.
2006 South East Mathematical Olympiad
Find the minimum value of real number $m$, such that inequality $$m(a^3+b^3+c^3) \ge 6(a^2+b^2+c^2)+1$$ holds for all positive real numbers $a,b,c$ where $a+b+c=1$.
2006 South East Mathematical Olympiad
(1) Find the number of positive integer solutions $(m,n,r)$ of the indeterminate equation $mn+nr+mr=2(m+n+r)$.

(2) Given an integer $k (k>1)$, prove that indeterminate equation $mn+nr+mr=k(m+n+r)$ has at least $3k+1$ positive integer solutions $(m,n,r)$.
2006 South East Mathematical Olympiad
Given a circle with its perimeter equal to $n$( $n \in N^*$), the least positive integer $P_n$ which satisfies the following condition is called the “number of the partitioned circle”: there are $P_n$ points ($A_1,A_2, \ldots ,A_{P_n}$) on the circle; For any integer $m$ ($1\le m\le n-1$), there always exist two points $A_i,A_j$ ($1\le i,j\le P_n$), such that the length of arc $A_iA_j$ is equal to $m$. Furthermore, all arcs between every two adjacent points $A_i,A_{i+1}$ ($1\le i\le P_n$, $A_{p_n+1}=A_1$) form a sequence $T_n=(a_1,a_2,,,a_{p_n})$ called the “sequence of the partitioned circle”. For example when $n=13$, the number of the partitioned circle $P_{13}$=4, the sequence of the partitioned circle $T_{13}=(1,3,2,7)$ or $(1,2,6,4)$. Determine the values of  $P_{21}$ and $P_{31}$, and find a possible solution of $T_{21}$ and $T_{31}$ respectively.
2005 South East Mathematical Olympiad
Let $a \in \mathbb{R}$ be a parameter.



(1) Prove that the curves of $y = x^2 + (a + 2)x - 2a + 1$ pass through a fixed point; also, the vertices of these parabolas all lie on the curve of a certain parabola.



(2) If the function $x^2 + (a + 2)x - 2a + 1 = 0$ has two distinct real roots, find the value range of the larger root.
2005 South East Mathematical Olympiad
Circle $C$ (with center $O$) does not have common point with line $l$. Draw $OP$ perpendicular to $l$, $P \in l$. Let $Q$ be a point on $l$ ($Q$ is different from $P$), $QA$ and $QB$ are tangent to circle $C$, and intersect the circle at $A$ and $B$ respectively. $AB$ intersects $OP$ at $K$. $PM$, $PN$ are perpendicular to $QB$, $QA$, respectively, $M \in QB$, $N \in QA$. Prove that segment $KP$ is bisected by line $MN$.