Navvye/Olympiad-Problems-CSV huggingface dataset:
2021 South East Mathematical Olympiad
Let $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$ be a set with $2n$ distinct elements, and $B_i\subseteq A$ for any $i=1,2,\cdots,m.$ If $\bigcup_{i=1}^m B_i=A,$ we say that the ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A.$ If $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A,$ and for any $i=1,2,\cdots,m$ and any $j=1,2,\cdots,n,$ $\{a_j,b_j\}$ is not a subset of $B_i,$ then we say that ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A$ without pairs. Define $a(m,n)$ as the number of the ordered $m-$coverings of $A,$ and $b(m,n)$ as the number of the ordered $m-$coverings of $A$ without pairs.

$(1)$ Calculate $a(m,n)$ and $b(m,n).$

$(2)$ Let $m\geq2,$ and there is at least one positive integer $n,$ such that $\dfrac{a(m,n)}{b(m,n)}\leq2021,$ Determine the greatest possible values of $m.$
2021 South East Mathematical Olympiad
Let $ABCD$ be a cyclic quadrilateral. The internal angle bisector of $\angle BAD$ and line $BC$ intersect at $E.$ $M$ is the midpoint of segment $AE.$ The exterior angle bisector of $\angle BCD$ and line $AD$ intersect at $F.$ The lines $MF$ and $AB$ intersect at $G.$ Prove that if $AB=2AD,$ then $MF=2MG.$
2021 South East Mathematical Olympiad
Determine all the pairs of positive odd integers $(a,b),$ such that $a,b>1$ and  $$7\varphi^2(a)-\varphi(ab)+11\varphi^2(b)=2(a^2+b^2),$$where $\varphi(n)$ is Euler's totient function.
2021 South East Mathematical Olympiad
A sequence $\{z_n\}$ satisfies that for any positive integer $i,$ $z_i\in\{0,1,\cdots,9\}$ and $z_i\equiv i-1 \pmod {10}.$ Suppose there is $2021$ non-negative reals $x_1,x_2,\cdots,x_{2021}$ such that for $k=1,2,\cdots,2021,$ $$\sum_{i=1}^kx_i\geq\sum_{i=1}^kz_i,\sum_{i=1}^kx_i\leq\sum_{i=1}^kz_i+\sum_{j=1}^{10}\dfrac{10-j}{50}z_{k+j}.$$Determine the least possible value of $\sum_{i=1}^{2021}x_i^2.$
2020 South East Mathematical Olympiad
Let $f(x)=a(3a+2c)x^2-2b(2a+c)x+b^2+(c+a)^2$ $(a,b,c\in R, a(3a+2c)\neq 0).$ If $$f(x)\leq 1$$for any real $x$, find the maximum of $|ab|.$
2020 South East Mathematical Olympiad
In a scalene triangle $\Delta ABC$, $AB<AC$, $PB$ and $PC$ are tangents of the circumcircle $(O)$ of $\Delta ABC$. A point $R$ lies on the arc $\widehat{AC}$(not containing $B$), $PR$ intersects $(O)$ again at $Q$. Suppose $I$ is the incenter of $\Delta ABC$, $ID \perp BC$ at $D$, $QD$ intersects $(O)$ again at $G$. A line passing through $I$ and perpendicular to $AI$ intersects $AB,AC$ at $M,N$, respectively.

Prove that, if $AR \parallel BC$, then $A,G,M,N$ are concyclic.
2020 South East Mathematical Olympiad
Given a polynomial $f(x)=x^{2020}+\sum_{i=0}^{2019} c_ix^i$, where $c_i \in \{ -1,0,1 \}$. Denote $N$ the number of positive integer roots of $f(x)=0$ (counting multiplicity). If $f(x)=0$ has no negative integer roots, find the maximum of $N$.
2020 South East Mathematical Olympiad
Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ .
2020 South East Mathematical Olympiad
Consider the set $I=\{ 1,2, \cdots, 2020 \}$. Let $W= \{w(a,b)=(a+b)+ab | a,b \in I \} \cap I$, $Y=\{y(a,b)=(a+b) \cdot ab | a,b \in I \} \cap I$ be its two subsets. Further, let $X= W \cap Y$.



(1) Find the sum of maximal and minimal elements in $X$.

(2) An element $n=y(a,b) (a \le b)$ in $Y$ is called excellent, if its representation is not unique (for instance, $20=y(1,5)=y(2,3)$). Find the number of excellent elements in $Y$.



Note(2) is only for Grade 11.
2020 South East Mathematical Olympiad
In a quadrilateral $ABCD$, $\angle ABC=\angle ADC <90^{\circ}$. The circle with diameter $AC$ intersects $BC$ and $CD$ again at $E,F$, respectively. $M$ is the midpoint of $BD$, and $AN \perp BD$ at $N$.

Prove that $M,N,E,F$ is concyclic.
2020 South East Mathematical Olympiad
Given any prime $p \ge 3$. Show that for all sufficient large positive integer $x$, at least one of $x+1,x+2,\cdots,x+\frac{p+3}{2}$ has a prime divisor greater than $p$.
2020 South East Mathematical Olympiad
Using a nozzle to paint each square in a $1 \times n$ stripe, when the nozzle is aiming at the $i$-th square, the square is painted black, and simultaneously, its left and right neighboring square (if exists) each has an independent probability of $\tfrac{1}{2}$ to be painted black.



In the optimal strategy (i.e. achieving least possible number of painting), the expectation of number of painting to paint all the squares black, is $T(n)$. Find the explicit formula of $T(n)$.
2020 South East Mathematical Olympiad
Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=2^n$ . Find the maximum possible value of positive integer $n$ .
2020 South East Mathematical Olympiad
In a scalene triangle $\Delta ABC$, $AB<AC$, $PB$ and $PC$ are tangents of the circumcircle $(O)$ of $\Delta ABC$. A point $R$ lies on the arc $\widehat{AC}$(not containing $B$), $PR$ intersects $(O)$ again at $Q$. Suppose $I$ is the incenter of $\Delta ABC$, $ID \perp BC$ at $D$, $QD$ intersects $(O)$ again at $G$. A line passing through $I$ and perpendicular to $AI$ intersects $AG,AC$ at $M,N$, respectively. $S$ is the midpoint of arc $\widehat{AR}$, and$SN$ intersects $(O)$ again at $T$.

Prove that, if $AR \parallel BC$, then $M,B,T$ are collinear.
2020 South East Mathematical Olympiad
Let $0\leq a_1\leq a_2\leq\hdots\leq a_{n-1}\leq a_n$ such that $a_1+a_2+\hdots+a_n=1$. Prove for any non-negative numbers $x_1,x_2,\hdots,x_n$, $y_1,y_2,\hdots,y_n$ the inequality

$$\left(\sum_{i=1}^n a_ix_i - \prod_{i=1}^n x_i^{a_i}\right)

\left(\sum_{i=1}^n a_iy_i - \prod_{i=1}^n y_i^{a_i}\right) \leq

a_n^2\left(n\sqrt{\sum_{i=1}^n x_i\sum_{i=1}^n y_i} - \sum_{i=1}^n\sqrt{x_i} \sum_{i=1}^n\sqrt{y_i}\right)^2.$$
2020 South East Mathematical Olympiad
Arrange all square-free positive integers in ascending order $a_1,a_2,a_3,\ldots,a_n,\ldots$. Prove that there are infinitely many positive integers $n$, such that $a_{n+1}-a_n=2020$.
2019 South East Mathematical Olympiad
Find the largest real number $k$, such that for any positive real numbers $a,b$,

$$(a+b)(ab+1)(b+1)\geq kab^2$$
2019 South East Mathematical Olympiad
Two circles $\Gamma_1$ and $\Gamma_2$ intersect at $A,B$. Points $C,D$ lie on $\Gamma_1$, points $E,F$ lie on $\Gamma_2$ such that $A,B$ lies on segments $CE,DF$ respectively and segments $CE,DF$ do not intersect. Let $CF$ meet $\Gamma_1,\Gamma_2$ again at $K,L$ respectively, and $DE$ meet $\Gamma_1,\Gamma_2$ at $M,N$ respectively. If the circumcircles of $\triangle ALM$ and $\triangle BKN$ are tangent, prove that the radii of these two circles are equal.
2019 South East Mathematical Olympiad
Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a function such that $f(ab)$ divides $\max \{f(a),b\}$ for any positive integers $a,b$. Must there exist infinitely many positive integers $k$ such that $f(k)=1$?
2019 South East Mathematical Olympiad
As the figure is shown, place a $2\times 5$ grid table in horizontal or vertical direction, and then remove arbitrary one $1\times 1$ square on its four corners. The eight different shapes consisting of the remaining nine small squares are called banners.

[asy]

defaultpen(linewidth(0.4)+fontsize(10));size(50);

pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9);

draw(B--C--H--J--N^^B--I^^D--N^^E--M^^F--L^^G--K);

draw(Aa--Ca--Ha--Ja--Aa^^Ba--Ia^^Da--Na^^Ea--Ma^^Fa--La^^Ga--Ka);

[/asy]

[asy]

defaultpen(linewidth(0.4)+fontsize(10));size(50);

pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9);

draw(B--Ca--Ea--M--N^^B--O^^C--E^^Aa--Ma^^Ba--Oa^^Da--N);

draw(L--Fa--Ha--J--L^^Ga--K^^P--I^^F--H^^Ja--La^^Pa--Ia);

[/asy]

Here is a fixed $9\times 18$ grid table. Find the number of ways to cover the grid table completely with 18 banners.
2019 South East Mathematical Olympiad
Let $S=\{1928,1929,1930,\cdots,1949\}.$ We call one of $S$’s subset $M$ is a red subset, if the sum of any two different elements of $M$ isn’t divided by $4.$ Let $x,y$ be the number of the red subsets of $S$ with $4$ and $5$ elements,respectively. Determine which of $x,y$ is greater and prove that.
2019 South East Mathematical Olympiad
Let $a,b,c$ be the lengths of the sides of a given triangle.If positive reals $x,y,z$ satisfy $x+y+z=1,$ find the maximum of $axy+byz+czx.$
2019 South East Mathematical Olympiad
Let $ABCD$ be a given convex quadrilateral in a plane. Prove that there exist a line with four different points $P,Q,R,S$ on it and a square $A’B’C’D’$ such that $P$ lies on both line $AB$ and $A’B’,$ $Q$ lies on both line $BC$ and $B’C’,$ $R$ lies on both line $CD$ and $C’D’,$ $S$ lies on both line $DA$ and $D’A’.$
2019 South East Mathematical Olympiad
For positive integer $x>1$, define set $S_x$ as $$S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\},$$where $v_p(n)$ is the power of prime divisor $p$ in positive integer $n.$ Let $f(x)$ be the sum of all the elements of $S_x$ when $x>1,$ and $f(1)=1.$

Let $m$ be a given positive integer, and the sequence $a_1,a_2,\cdots,a_n,\cdots$ satisfy that for any positive integer $n>m,$ $a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}.$ Prove that

(1)there exists constant $A,B(0<A<1),$ such that when positive integer $x$ has at least two different prime divisors, $f(x)<Ax+B$ holds;

(2)there exists positive integer $Q$, such that for any positive integer $n,a_n<Q.$
2019 South East Mathematical Olympiad
Let $[a]$ represent the largest integer less than or equal to $a$, for any real number $a$. Let $\{a\} = a - [a]$.



Are there positive integers $m,n$ and $n+1$ real numbers $x_0,x_1,\hdots,x_n$ such that $x_0=428$, $x_n=1928$, $\frac{x_{k+1}}{10} = \left[\frac{x_k}{10}\right] + m + \left\{\frac{x_k}{5}\right\}$ holds?



Justify your answer.
2019 South East Mathematical Olympiad
$ABCD$ is a parallelogram with $\angle BAD \neq 90$. Circle centered at $A$ radius $BA$ denoted as $\omega _1$ intersects the extended side of $AB,CB$ at points $E,F$ respectively. Suppose the circle centered at $D$ with radius $DA$, denoted as $\omega _2$, intersects $AD,CD$ at points $M,N$ respectively. Suppose $EN,FM$ intersects at $G$, $AG$ intersects $ME$ at point $T$. $MF$ intersects $\omega _1$ at $Q \neq F$, and $EN$ intersects $\omega _2$ at $P \neq N$. Prove that $G,P,T,Q$ concyclic.
2019 South East Mathematical Olympiad
$n$ symbols line up in a row, numbered as $1,2,...,n$ from left to right. Delete every symbol with squared numbers. Renumber the rest from left to right. Repeat the process until all $n$ symbols are deleted. Let $f(n)$ be the initial number of the last symbol deleted. Find $f(n)$ in terms of $n$ and find $f(2019)$.
2019 South East Mathematical Olympiad
Let $X$ be a $5\times 5$ matrix with each entry be $0$ or $1$. Let $x_{i,j}$ be the $(i,j)$-th entry of $X$ ($i,j=1,2,\hdots,5$). Consider all the $24$ ordered sequence in the rows, columns and diagonals of $X$ in the following:

\begin{align*}

&(x_{i,1}, x_{i,2},\hdots,x_{i,5}),\ (x_{i,5},x_{i,4},\hdots,x_{i,1}),\ (i=1,2,\hdots,5) \\
&(x_{1,j}, x_{2,j},\hdots,x_{5,j}),\ (x_{5,j},x_{4,j},\hdots,x_{1,j}),\ (j=1,2,\hdots,5) \\
&(x_{1,1},x_{2,2},\hdots,x_{5,5}),\ (x_{5,5},x_{4,4},\hdots,x_{1,1}) \\
&(x_{1,5},x_{2,4},\hdots,x_{5,1}),\ (x_{5,1},x_{4,2},\hdots,x_{1,5})

\end{align*}Suppose that all of the sequences are different. Find all the possible values of the sum of all entries in $X$.
2019 South East Mathematical Olympiad
For positive integer n, define $a_n$ as the number of the triangles with integer length of every side and the length of the longest side being $2n.$

(1) Find $a_n$ in terms of $n;$

(2)If the sequence $\{ b_n\}$ satisfying for any positive integer $n,$ $\sum_{k=1}^n(-1)^{n-k}\binom {n}{k} b_k=a_n.$ Find the number of positive integer $n$ satisfying that $b_n\leq 2019a_n.$
2019 South East Mathematical Olympiad
In $\triangle ABC$, $AB>AC$, the bisectors of $\angle ABC, \angle ACB$ meet sides $AC,AB$ at $D,E$ respectively. The tangent at $A$ to the circumcircle of $\triangle ABC$ intersects $ED$ extended at $P$. Suppose that $AP=BC$. Prove that $BD\parallel CP$.
2019 South East Mathematical Olympiad
Amy and Bob choose numbers from $0,1,2,\cdots,81$ in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all $82$ numbers are chosen, let $A$ be the sum of all the numbers Amy chooses, and let $B$ be the sum of all the numbers Bob chooses. During the process, Amy tries to make $\gcd(A,B)$ as great as possible, and Bob tries to make $\gcd(A,B)$ as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine $\gcd(A,B)$ when all $82$ numbers are chosen.
2019 South East Mathematical Olympiad
For positive integer $x>1$, define set $S_x$ as $$S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\},$$where $v_p(n)$ is the power of prime divisor $p$ in positive integer $n.$ Let $f(x)$ be the sum of all the elements of $S_x$ when $x>1,$ and $f(1)=1.$

Let $m$ be a given positive integer, and the sequence $a_1,a_2,\cdots,a_n,\cdots$ satisfy that for any positive integer $n>m,$ $a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}.$ Prove that

(1)there exists constant $A,B(0<A<1),$ such that when positive integer $x$ has at least two different prime divisors, $f(x)<Ax+B$ holds;

(2)there exists positive integer $N,l$, such that for any positive integer $n\geq N ,a_{n+l}=a_n$ holds.
2018 South East Mathematical Olympiad
Assume $c$ is a real number. If there exists $x\in[1,2]$ such that $\max\left\{\left |x+\frac cx\right |, \left |x+\frac cx + 2\right |\right\}\geq 5$, please find the value range of $c$.
2018 South East Mathematical Olympiad
In a Cartesian plane, if both horizontal coordinate and vertical coordinate of a point are rational numbers, we call the point rational point. Otherwise, we call it irrational point. Consider an arbitrary regular pentagon on the Cartesian plane. Please compare the number of rational point and the number of irrational point among the five vertices of the pentagon. Prove your conclusion.
2018 South East Mathematical Olympiad
Let $O$ be the circumcenter of acute $\triangle ABC$($AB<AC$), the angle bisector of $\angle BAC$ meets $BC$ at $T$ and $M$ is the midpoint of $AT$. Point $P$ lies inside $\triangle ABC$ such that $PB\perp PC$. $D,E$ distinct from $P$ lies on the perpendicular to $AP$ through $P$ such that $BD=BP, CE=CP$. If $AO$ bisects segment $DE$, prove that $AO$ is tangent to the circumcircle of $\triangle AMP$.
2018 South East Mathematical Olympiad
Does there exist a set $A\subseteq\mathbb{N}^*$ such that for any positive integer $n$, $A\cap\{n,2n,3n,...,15n\}$ contains exactly one element? Please prove your conclusion.
2018 South East Mathematical Olympiad
Let $\{a_n\}$ be a nonnegative real sequence. Define

$$X_k = \sum_{i=1}^{2^k}a_i, Y_k = \sum_{i=1}^{2^k}\left\lfloor \frac{2^k}{i}\right\rfloor a_i, k=0,1,2,...$$Prove that $X_n\le Y_n - \sum_{i=0}^{n-1} Y_i \le \sum_{i=0}^n X_i$ for all positive integer $n$. Here $\lfloor\alpha\rfloor$ denotes the largest integer that does not exceed $\alpha$.
2018 South East Mathematical Olympiad
In the isosceles triangle $ABC$ with $AB=AC$, the center of $\odot O$ is the midpoint of the side $BC$, and $AB,AC$ are tangent to the circle at points $E,F$ respectively. Point $G$ is on $\odot O$ with $\angle AGE = 90^{\circ}$. A tangent line of $\odot O$ passes through $G$, and meets $AC$ at $K$. Prove that line $BK$ bisects $EF$.
2018 South East Mathematical Olympiad
There are $24$ participants attended a meeting. Each two of them shook hands once or not. A total of $216$ handshakes occured in the meeting. For any two participants who have shaken hands, at most $10$ among the rest $22$ participants have shaken hands with exactly one of these two persons. Define a friend circle to be a group of $3$ participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.
2018 South East Mathematical Olympiad
Given a positive integer $m$. Let

$$A_l = (4l+1)(4l+2)...(4(5^m+1)l)$$for any positive integer $l$. Prove that there exist infinite number of positive integer $l$ which

$$5^{5^ml}\mid A_l\text{ and } 5^{5^ml+1}\nmid A_l$$and find the minimum value of $l$ satisfying the above condition.
2018 South East Mathematical Olympiad
Suppose that $a$ is real number. Sequence $a_1,a_2,a_3,....$ satisfies

$$a_1=a, a_{n+1} = 

\begin{cases}

a_n - \frac{1}{a_n}, & a_n\ne 0 \\
0, & a_n=0

\end{cases}

(n=1,2,3,..)$$Find all possible values of $a$ such that $|a_n|<1$ for all positive integer $n$.
2018 South East Mathematical Olympiad
Let $O$ be the circumcenter of $\triangle ABC$, where $\angle ABC> 90^{\circ}$ and $M$ is the midpoint of $BC$. Point $P$ lies inside $\triangle ABC$ such that $PB\perp PC$. $D,E$ distinct from $P$ lies on the perpendicular to $AP$ through $P$ such that $BD=BP, CE=CP$. If quadrilateral $ADOE$ is a parallelogram, prove that $$\angle OPE = \angle AMB.$$
2018 South East Mathematical Olympiad
Does there exist a set $A\subseteq\mathbb{N}^*$ such that for any positive integer $n$, $A\cap\{n,2n,3n,...,15n\}$ contains exactly one element and there exists infinitely many positive integer $m$ such that $\{m,m+2018\}\subset A$? Please prove your conclusion.
2018 South East Mathematical Olympiad
Assume integer $m \geq 2.$ There are $3m$ people in a meeting, any two of them either shake hands with each other once or not.We call the meeting "$n$-interesting", only if there exists $n(n\leq 3m-1)$ people of them, the time everyone of whom shakes hands with other $3m-1$ people is exactly $1,2,\cdots,n,$ respectively. If in any "$n$-interesting" meeting, there exists $3$ people of them who shake hands with each other, find the minimum value of $n.$
2018 South East Mathematical Olympiad
For positive integers $m,n,$ define $f(m,n)$ as the number of ordered triples $(x,y,z)$ of integers such that

$$ 

\begin{cases}

xyz=x+y+z+m, \\
\max\{|x|,|y|,|z|\} \leq n

\end{cases}

$$Does there exist positive integers $m,n,$ such that $f(m,n)=2018?$ Please prove your conclusion.
2018 South East Mathematical Olympiad
Given a positive real $C \geq 1$ and a sequence $a_1, a_2, a_3, \cdots$ satisfying for any positive integer $n,$ $a_n \geq 0$

and for any real $x \geq 1$,

$$\left|x\lg x-\sum_{k=1}^{[x]}\left[\frac{x}{k}\right]a_k \right| \leq Cx,$$where $[x]$ is defined as the largest integer that does not exceed $x$. Prove that for any real $y \geq 1$,

$$\sum_{k=1}^{[y]}a_k < 3Cy.$$
2016 South East Mathematical Olympiad
The sequence $(a_n)$ is defined by $a_1=1,a_2=\frac{1}{2}$,$$n(n+1)

a_{n+1}a_{n}+na_{n}a_{n-1}=(n+1)^2a_{n+1}a_{n-1}(n\ge 2).$$Prove that $$\frac{2}{n+1}<\sqrt[n]{a_n}<\frac{1}{\sqrt{n}}(n\ge 3).$$
2016 South East Mathematical Olympiad
Suppose $PAB$ and $PCD$ are two secants of circle $O$. Lines $AD \cap BC=Q$. Point $T$ lie on segment $BQ$ and point $K$ is intersection of segment $PT$ with circle $O$, $S=QK\cap PA$

Given that $ST \parallel PQ$, prove that $B,S,K,T$ lie on a circle.
2016 South East Mathematical Olympiad
Given any integer $n\geq 3$. A finite series is called $n$-series if it satisfies the following two conditions

$1)$ It has at least $3$ terms and each term of it belongs to $\{ 1,2,...,n\}$

$2)$ If series has $m$ terms $a_1,a_2,...,a_m$ then $(a_{k+1}-a_k)(a_{k+2}-a_k)<0$ for all $k=1,2,...,m-2$



How many $n$-series are there $?$
2016 South East Mathematical Olympiad
For any four points on a plane, if the areas of four triangles formed are different positive integer and six distances between those four points are also six different positive integers, then the convex closure of $4$ points is called a "lotus design."

(1) Construct an example of "lotus design". Also what are areas and distances in your example?

(2) Prove that there are infinitely many "lotus design" which are not similar.
2016 South East Mathematical Olympiad
Let $n$ is positive integer, $D_n$ is a set of all positive divisor of $n$ and $f(n)=\sum_{d\in D_n}{\frac{1}{1+d}}$

Prove that for all positive integer $m$, $\sum_{i=1}^{m}{f(i)} <m$
2016 South East Mathematical Olympiad
Toss the coin $n$ times, assume that each time, only appear only head or tail

Let $a(n)$ denote number of way that head appear in multiple of $3$ times among $n$ times

Let $b(n)$ denote numbe of way that head appear in multiple of $6$ times among $n$ times

$(1)$ Find $a(2016)$ and $b(2016)$

$(2)$ Find the number of positive integer $n\leq 2016$ that $2b(n)-a(n)\geq 0$
2016 South East Mathematical Olympiad
$I$ is incenter of $\triangle{ABC}$. The incircle touches $BC,CA,AB$ at $D,E,F$, respectively .

Let $M,N,K=BI,CI,DI \cap EF$ respectively and $BN\cap CM=P,AK\cap BC=G$.

Point $Q$ is intersection of the perpendicular line to $PG$ through $I$ and the perpendicular line to $PB$ through $P$.

Prove that $BI$ bisect segment $PQ$.
2016 South East Mathematical Olympiad
Let $\{ a_n\}$ be a series consisting of positive integers such that $n^2 \mid \sum_{i=1}^{n}{a_i}$ and $a_n\leq (n+2016)^2$ for all $n\geq 2016$.

Define $b_n=a_{n+1}-a_n$. Prove that the series $\{ b_n\}$ is eventually constant.
2016 South East Mathematical Olympiad
Let $n$ be positive integer,$x_1,x_2,\cdots,x_n$ be  positive real numbers such that $x_1x_2\cdots x_n=1 $ . Prove that$$\sum\limits_{i = 1}^{n}x_i\sqrt{x^2_1+x^2_2+\cdots x^2_i}\ge\frac{n+1}{2}\sqrt{n}$$
2016 South East Mathematical Olympiad
A substitute teacher lead a groop of students to go for a trip. The teacher who in charge of the groop of the students told the substitude teacher that there are two students who always lie, and the others always tell the truth. But the substitude teacher don't know who are the two students always lie. They get lost in a forest. Finally the are in a crossroad which has four roads. The substitute teacher knows that their camp is on one road, and the distence is $20$ minutes' walk. The students have to go to the camp before it gets dark.

$(1)$ If there are $8$ students, and $60$ minutes before it gets dark, give a plan that all students can get back to the camp.

$(2)$ If there are $4$ students, and $100$ minutes before it gets dark, is there a plan that all students can get back to the camp?
2016 South East Mathematical Olympiad
Let a constant $\alpha$ as $0<\alpha<1$, prove that:

$(1)$ There exist a constant $C(\alpha)$ which is only depend on $\alpha$ such that for every $x\ge 0$, $\ln(1+x)\le C(\alpha)x^\alpha$.

$(2)$ For every two complex numbers $z_1,z_2$, $|\ln|\frac{z_1}{z_2}||\le C(\alpha)\left(|\frac{z_1-z_2}{z_2}|^\alpha+|\frac{z_2-z_1}{z_1}|^\alpha\right)$.
2016 South East Mathematical Olympiad
Let $A=\{a^3+b^3+c^3-3abc|a,b,c\in\mathbb{N}\}$, $B=\{(a+b-c)(b+c-a)(c+a-b)|a,b,c\in\mathbb{N}\}$, $P=\{n|n\in A\cap B,1\le n\le 2016\}$, find the value of $|P|$.
2015 South East Mathematical Olympiad
Suppose that the sequence $\{a_n\}$ satisfy $a_1=1$ and $a_{2k}=a_{2k-1}+a_k, \quad a_{2k+1}=a_{2k}$ for $k=1,2, \ldots$

Prove that $a_{2^n}< 2^{\frac{n^2}{2}}$ for any integer $n \geq 3$.
2015 South East Mathematical Olympiad
Let $I$ be the incenter of $\triangle ABC$ with $AB>AC$. Let $\Gamma$ be the circle with diameter $AI$. The circumcircle of $\triangle ABC$ intersects $\Gamma$ at points $A,D$, with point $D$ lying on $\overarc{AC}$ (not containing $B$). Let the line passing through $A$ and parallel to $BC$ intersect $\Gamma$ at points $A,E$. If $DI$ is the angle bisector of $\angle CDE$, and $\angle ABC = 33^{\circ}$, find the value of $\angle BAC$.