Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 China Girls Math Olympiad
Let $DFGE$ be a cyclic quadrilateral. Line $DF$ intersects $EG$ at $C,$ and line $FE$ intersects $DG$ at $H.$ $J$ is the midpoint of $FG.$ The line $\ell$ is the reflection of the line $DE$ in $CH,$ and it intersects line $GF$ at $I.$

Prove that $C,J,H,I$ are concyclic.
2019 China Girls Math Olympiad
For a tournament with $8$ vertices, if from any vertex it is impossible to follow a route to return to itself, we call the graph a good graph. Otherwise, we call it a bad graph. Prove that

$(1)$ there exists a tournament with $8$ vertices such that after changing the orientation of any at most $7$ edges of the tournament, the graph is always abad graph;

$(2)$ for any tournament with $8$ vertices, one can change the orientation of at most $8$ edges of the tournament to get a good graph.

(A tournament is a complete graph with directed edges.)
2018 China Girls Math Olympiad
Let $a\le 1$ be a real number. Sequence $\{x_n\}$ satisfies $x_0=0, x_{n+1}= 1-a\cdot e^{x_n}$, for all $n\ge 1$, where $e$ is the natural logarithm. Prove that for any natural $n$, $x_n\ge 0$.
2018 China Girls Math Olympiad
Points $D,E$ lie on segments $AB,AC$ of $\triangle ABC$ such that $DE\parallel BC$. Let $O_1,O_2$ be the circumcenters of $\triangle ABE, \triangle ACD$ respectively. Line $O_1O _2$ meets $AC$ at $P$, and $AB$ at $Q$. Let $O$ be the circumcenter of $\triangle APQ$, and $M$ be the intersection of $AO$ extended and $BC$. Prove that $M$ is the midpoint of $BC$.
2018 China Girls Math Olympiad
Given a real sequence $\left \{ x_n \right \}_{n=1}^{\infty}$ with $x_1^2 = 1$. Prove that for each integer $n \ge 2$, $$\sum_{i|n}\sum_{j|n}\frac{x_ix_j}{\textup{lcm} \left ( i,j \right )} \ge \prod_{\mbox{\tiny$\begin{array}{c}

p \: \textup{is prime} \\ p|n \end{array}$} }\left ( 1-\frac{1}{p} \right ). $$
2018 China Girls Math Olympiad
There're $n$ students whose names are different from each other. Everyone has $n-1$ envelopes initially with the others' name and address written on them respectively. Everyone also has at least one greeting card with her name signed on it. Everyday precisely a student encloses a greeting card (which can be the one received before) with an envelope (the name on the card and the name on envelope cannot be the same) and post it to the appointed student by a same day delivery.

Prove that when no one can post the greeting cards in this way any more:

(i) Everyone still has at least one card;

(ii) If there exist $k$ students $p_1, p_2, \cdots, p_k$ so that $p_i$ never post a card to $p_{i+1}$, where $i = 1,2, \cdots, k$ and $p_{k+1} = p_1$, then these $k$ students have prepared the same number of greeting cards initially.
2018 China Girls Math Olympiad
Let $\omega \in \mathbb{C}$, and $\left |  \omega  \right | = 1$. Find the maximum length of $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$.
2018 China Girls Math Olympiad
Given $k \in \mathbb{N}^+$.  A sequence of subset of the integer set $\mathbb{Z} \supseteq  I_1 \supseteq  I_2 \supseteq \cdots \supseteq I_k$ is called a $k-chain$ if for each $1 \le i \le k$ we have

(i) $168 \in I_i$;

(ii) $\forall x, y \in I_i$, we have $x-y \in I_i$.

Determine the number of $k-chain$ in total.
2018 China Girls Math Olympiad
Given $2018 \times 4$ grids and tint them with red and blue. So that each row and each column has the same number of red and blue grids, respectively. Suppose there're $M$ ways to tint the grids with  the mentioned requirement. Determine $M \pmod {2018}$.
2018 China Girls Math Olympiad
Let $I$ be the incenter of triangle $ABC$. The tangent point of $\odot I$ on $AB,AC$ is $D,E$, respectively. Let $BI \cap AC = F$, $CI \cap AB = G$, $DE \cap BI = M$, $DE \cap CI = N$, $DE \cap FG = P$, $BC \cap IP = Q$. Prove that $BC = 2MN$ is equivalent to $IQ = 2IP$.
2017 China Girls Math Olympiad
(1) Find all positive integer $n$ such that for any odd integer $a$, we have $4\mid a^n-1$

(2) Find all positive integer $n$ such that for any odd integer $a$, we have $2^{2017}\mid a^n-1$
2017 China Girls Math Olympiad
Given quadrilateral $ABCD$ such that $\angle BAD+2 \angle BCD=180 ^ \circ .$

Let $E$ be the intersection of $BD$ and the internal bisector of  $\angle BAD$.

The perpendicular bisector of $AE$ intersects $CB,CD$ at $X,Y,$ respectively.

Prove that $A,C,X,Y$ are concyclic.
2017 China Girls Math Olympiad
Given $a_i\ge 0,x_i\in\mathbb{R},(i=1,2,\ldots,n)$. Prove that

$$((1-\sum_{i=1}^n a_i\cos x_i)^2+(1-\sum_{i=1}^n a_i\sin x_i)^2)^2\ge 4(1-\sum_{i=1}^n a_i)^3$$
2017 China Girls Math Olympiad
Partition $\frac1{2002},\frac1{2003},\frac1{2004},\ldots,\frac{1}{2017}$ into two groups. Define $A$ the sum of the numbers in the first group, and $B$ the sum of the numbers in the second group. Find the partition such that $|A-B|$ attains it minimum and explains the reason.
2017 China Girls Math Olympiad
Let $0=x_0<x_1<\cdots<x_n=1$ .Find the largest real number$ C$ such  that for any positive integer $ n $ ,  we have $$\sum_{k=1}^n  x^2_k (x_k - x_{k-1})>C$$
2017 China Girls Math Olympiad
Given a finite set $X$, two positive integers $n,k$, and a map $f:X\to X$. Define $f^{(1)}(x)=f(x),f^{(i+1)}(x)=f^{(i)}(x)$,$i=1,2,3,\ldots$. It is known that for any $x\in X$,$f^{(n)}(x)=x$.

Define $m_j$ the number of $x\in X$ satisfying $f^{(j)}(x)=x$.

Prove that:

(1)$\frac{1}n \sum_{j=1}^n  m_j\sin {\frac{2kj\pi}{n}}=0$

(2)$\frac{1}n \sum_{j=1}^n  m_j\cos {\frac{2kj\pi}{n}}$ is a non-negative integer.
2017 China Girls Math Olympiad
This is a very classical problem.

Let the $ABCD$ be a cyclic quadrilateral with circumcircle $\omega_1$.Lines $AC$ and $BD$ intersect at point $E$,and lines $AD$,$BC$ intersect at point $F$.Circle $\omega_2$ is tangent to segments $EB,EC$ at points $M,N$ respectively,and intersects with circle $\omega_1$ at points $Q,R$.Lines $BC,AD$ intersect line $MN$ at $S,T$ respectively.Show that $Q,R,S,T$ are concyclic.
2017 China Girls Math Olympiad
Let $n$ be a fixed positive integer. Let

$$A=\begin{bmatrix} a_{11} & a_{12} & \cdots &a_{1n}  \\ a_{21} & a_{22} & \cdots &a_{2n}  \\ \vdots & \vdots & \cdots & \vdots \\ a_{n1} & a_{n2} & \cdots &a_{nn} \end{bmatrix}\quad

\text{and} \quad B=\begin{bmatrix} b_{11} & b_{12} & \cdots &b_{1n}  \\ b_{21} & b_{22} & \cdots &b_{2n}  \\ \vdots & \vdots & \cdots & \vdots \\ b_{n1} & b_{n2} & \cdots &b_{nn} \end{bmatrix}\quad$$be two $n\times n$ tables such that $\{a_{ij}|1\le i,j\le n\}=\{b_{ij}|1\le i,j\le n\}=\{k\in N^*|1\le k\le n^2\}$.

One can perform the following operation on table $A$: Choose $2$ numbers in the same row or in the same column of $A$, interchange these $2$ numbers, and leave the remaining $n^2-2$ numbers unchanged. This operation is called a transposition of $A$.

Find, with proof, the smallest positive integer $m$ such that for any tables $A$ and $B$, one can perform at most $m$ transpositions such that the resulting table of $A$ is $B$.
2016 China Girls Math Olympiad
Let $n\ge 3$ be an integer. Put $n^2$ cards, each labelled $1,2,\ldots ,n^2$ respectively, in any order into $n$ empty boxes such that there are exactly $n$ cards in each box. One can perform the following operation: one first selects $2$ boxes, takes out any $2$ cards from each of the selected boxes, and then return the cards to the other selected box. Prove that, for any initial order of the $n^2$ cards in the boxes, one can perform the operation finitely many times such that the labelled numbers in each box are consecutive integers.
2016 China Girls Math Olympiad
In $\triangle ABC, BC=a, CA=b, AB=c,$ and $\Gamma$ is its circumcircle.

$(1)$ Determine a necessary and sufficient condition on $a,b$ and $c$ if there exists a unique point $P(P\neq B, P\neq C)$ on the arc $BC$ of $\Gamma$ not passing through point $A$ such that $PA=PB+PC$.

$(2)$ Let $P$ be the unique point stated in $(1)$. If $AP$ bisects $BC$, prove that $\angle BAC<60^{\circ}$.
2016 China Girls Math Olympiad
Let  $m$ and $n$ are relatively prime integers and $m>1,n>1$. Show that:There are positive integers $a,b,c$ such that $m^a=1+n^bc$ , and  $n$ and $c$ are relatively prime.
2016 China Girls Math Olympiad
Let $n$ is a positive integers ,$a_1,a_2,\cdots,a_n\in\{0,1,\cdots,n\}$ . For the integer $j$ $(1\le j\le n)$ ,define $b_j$ is the number of elements in the set $\{i|i\in\{1,\cdots,n\},a_i\ge j\}$ .For example ：When $n=3$ ,if $a_1=1,a_2=2,a_3=1$ ,then $b_1=3,b_2=1,b_3=0$ .

$(1)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^2\ge \sum_{i=1}^{n}(i+b_i)^2.$$$(2)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^k\ge \sum_{i=1}^{n}(i+b_i)^k,$$for the integer $k\ge 3.$
2016 China Girls Math Olympiad
Define a sequence $\{a_n\}$  by$$S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots).$$Where $S_n$ the sum of first $n$ terms of sequence $\{a_n\}$.

For any positive integer $n$ ,prove that$$a_{n}\ge \frac{4}{\sqrt{9n+7}}.$$
2016 China Girls Math Olympiad
Find the greatest positive integer $m$, such that one of the $4$ letters $C,G,M,O$ can be placed in each cell of a table with $m$ rows and $8$ columns, and has the following property: For any two distinct rows in the table, there exists at most one column, such that the entries of these two rows in such a column are the same letter.
2016 China Girls Math Olympiad
In acute triangle $ABC, AB<AC$, $I$ is its incenter, $D$ is the foot of perpendicular from $I$ to $BC$, altitude $AH$ meets $BI,CI$ at $P,Q$ respectively. Let $O$ be the circumcenter of $\triangle IPQ$, extend $AO$ to meet $BC$ at $L$. Circumcircle of $\triangle AIL$ meets $BC$ again at $N$. Prove that $\frac{BD}{CD}=\frac{BN}{CN}$.
2016 China Girls Math Olympiad
Let $\mathbb{Q}$ be the set of rational numbers, $\mathbb{Z}$ be the set of integers. On the coordinate plane, given positive integer $m$, define $$A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}.$$For segment $MN$, define $f_m(MN)$ as the number of points on segment $MN$ belonging to set $A_m$.



Find the smallest real number $\lambda$, such that for any line $l$ on the coordinate plane, there exists a constant $\beta (l)$ related to $l$, satisfying: for any two points $M,N$ on $l$, $$f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l)$$
2015 China Girls Math Olympiad
Let $\triangle ABC$ be an acute-angled triangle with $AB>AC$, $O$ be its circumcenter and $D$ the midpoint of side $BC$. The circle with diameter $AD$ meets sides $AB,AC$ again at points $E,F$ respectively. The line passing through $D$ parallel to $AO$ meets $EF$ at $M$. Show that $EM=MF$.
2015 China Girls Math Olympiad
Let $a\in(0,1)$ ,$f(x)=ax^3+(1-4a)x^2+(5a-1)x-5a+3 $ , $g(x)=(1-a)x^3-x^2+(2-a)x-3a-1 $.

Prove that:For any real number $x$ ,at least one of $|f(x)|$  and $|g(x)|$ not less than $a+1$.
2015 China Girls Math Olympiad
In a $12\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\times 4$ and $4\times 3$ rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares.
2015 China Girls Math Olympiad
Let $g(n)$ be the greatest common divisor of $n$ and $2015$. Find the number of triples $(a,b,c)$ which satisfies the following two conditions:

$1)$ $a,b,c \in$ {$1,2,...,2015$};

$2)$ $g(a),g(b),g(c),g(a+b),g(b+c),g(c+a),g(a+b+c)$ are pairwise distinct.
2015 China Girls Math Olympiad
Determine the number of distinct right-angled triangles such that its three sides are of integral lengths, and its area is $999$ times of its perimeter.

(Congruent triangles are considered identical.)
2015 China Girls Math Olympiad
Let $\Gamma_1$ and $\Gamma_2$ be two non-overlapping circles. $A,C$ are on $\Gamma_1$ and $B,D$ are on $\Gamma_2$ such that $AB$ is an external common tangent to the two circles, and $CD$ is an internal common tangent to the two circles. $AC$ and $BD$ meet at $E$. $F$ is a point on $\Gamma_1$, the tangent line to $\Gamma_1$ at $F$ meets the perpendicular bisector of $EF$ at $M$. $MG$ is a line tangent to $\Gamma_2$ at $G$. Prove that $MF=MG$.
2015 China Girls Math Olympiad
Let $x_1,x_2,\cdots,x_n \in(0,1)$ , $n\geq2$. Prove that$$\frac{\sqrt{1-x_1}}{x_1}+\frac{\sqrt{1-x_2}}{x_2}+\cdots+\frac{\sqrt{1-x_n}}{x_n}<\frac{\sqrt{n-1}}{x_1 x_2 \cdots x_n}.$$
2015 China Girls Math Olympiad
Let $n\geq 2$ be a given integer. Initially, we write $n$ sets on the blackboard and do a sequence of moves as follows: choose two sets $A$ and $B$ on the blackboard such that none of them is a subset of the other, and replace $A$ and $B$ by $A\cap B$ and $A\cup B$. This is called a $\textit{move}$.

Find the maximum number of moves in a sequence for all possible initial sets.
2014 China Girls Math Olympiad
In the figure of 

$\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$.

The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$,

and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$.

If $DE \parallel O_1A$, prove that $DC \perp CO_2$.
2014 China Girls Math Olympiad
Let $x_1,x_2,\ldots,x_n $ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $ be a permutation of $1,2,\ldots,n$.

Find the maximum and  minimum of  $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor $ is the largest integer not greater than  $x$).
2014 China Girls Math Olympiad
There are $ n$  students; each student knows exactly $d $  girl students and $d $ boy students ("knowing" is a symmetric relation). Find all pairs $ (n,d) $  of integers .
2014 China Girls Math Olympiad
For an integer $m\geq 4,$ let $T_{m}$ denote the number of sequences $a_{1},\dots,a_{m}$ such that the following conditions hold:

(1) For all $i=1,2,\dots,m$ we have $a_{i}\in \{1,2,3,4\}$

(2) $a_{1} = a_{m} = 1$ and $a_{2}\neq 1$

(3) For all $i=3,4\cdots, m, a_{i}\neq a_{i-1}, a_{i}\neq a_{i-2}.$



Prove that there exists a geometric sequence of positive integers $\{g_{n}\}$ such that for $n\geq 4$ we have that $$ g_{n} - 2\sqrt{g_{n}} < T_{n} < g_{n} + 2\sqrt{g_{n}}.$$
2014 China Girls Math Olympiad
Let $a$ be a positive integer, but not a perfect square; $r$ is a real root of the equation $x^3-2ax+1=0$. Prove that  $ r+\sqrt{a}$ is an irrational number.
2014 China Girls Math Olympiad
In acute triangle $ABC$, $AB > AC$.

$D$ and $E$ are the midpoints of $AB$, $AC$ respectively.

The circumcircle of $ADE$ intersects the circumcircle of $BCE$ again at $P$.

The circumcircle of $ADE$ intersects the circumcircle $BCD$ again at $Q$.

Prove that $AP = AQ$.
2014 China Girls Math Olympiad
Given a finite nonempty set $X$ with real values, let $f(X) = \frac{1}{|X|} \displaystyle\sum\limits_{a\in X} a$, where $\left\lvert X \right\rvert$ denotes the cardinality of $X$.  For ordered pairs of sets $(A,B)$ such that $A\cup B = \{1, 2, \dots , 100\}$ and $A\cap B = \emptyset$ where $1\leq |A| \leq 98$, select some $p\in B$, and let $A_{p} = A\cup \{p\}$ and $B_{p} = B - \{p\}.$ Over all such $(A,B)$ and $p\in B$ determine the maximum possible value of $(f(A_{p})-f(A))(f(B_{p})-f(B)).$
2014 China Girls Math Olympiad
Let $n$ be a positive integer, and set $S$ be the set of all integers in $\{1,2,\dots,n\}$ which are relatively prime to $n$.

Set $S_1 = S \cap \left(0, \frac n3 \right]$, $S_2 = S \cap \left( \frac n3, \frac {2n}3 \right]$, $S_3 = S \cap \left( \frac{2n}{3}, n \right]$.

If the cardinality of $S$ is a multiple of $3$, prove that $S_1$, $S_2$, $S_3$ have the same cardinality.
2013 China Girls Math Olympiad
Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$, where $0<t<1$. Prove that the area of any triangle inside the region $A$, with two vertices $P(t,t^2)$ and $Q(1,0)$, does not exceed $\frac{1}{4}.$
2013 China Girls Math Olympiad
As shown in the figure below, $ABCD$ is a trapezoid, $AB \parallel CD$.  The sides $DA$, $AB$, $BC$ are tangent to $\odot O_1$ and $AB$ touches $\odot O_1$ at $P$.  The sides $BC$, $CD$, $DA$ are tangent to $\odot O_2$, and $CD$ touches $\odot O_2$ at $Q$.  Prove that the lines $AC$, $BD$, $PQ$ meet at the same point.



[asy]

size(200);

defaultpen(linewidth(0.8)+fontsize(10pt));

pair A=origin,B=(1,-7),C=(30,-15),D=(26,6);

pair bisA=bisectorpoint(B,A,D),bisB=bisectorpoint(A,B,C),bisC=bisectorpoint(B,C,D),bisD=bisectorpoint(C,D,A);

path bA=A--(bisA+100*(bisA-A)),bB=B--(bisB+100*(bisB-B)),bC=C--(bisC+100*(bisC-C)),bD=D--(bisD+100*(bisD-D));

pair O1=intersectionpoint(bA,bB),O2=intersectionpoint(bC,bD);

dot(O1^^O2,linewidth(2));

pair h1=foot(O1,A,B),h2=foot(O2,C,D);

real r1=abs(O1-h1),r2=abs(O2-h2);

draw(circle(O1,r1)^^circle(O2,r2));

draw(A--B--C--D--cycle);

draw(A--C^^B--D^^h1--h2);

label("$A$",A,NW);

label("$B$",B,SW);

label("$C$",C,dir(350));

label("$D$",D,dir(350));

label("$P$",h1,dir(200));

label("$Q$",h2,dir(350));

label("$O_1$",O1,dir(150));

label("$O_2$",O2,dir(300));

[/asy]
2013 China Girls Math Olympiad
In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$.
2013 China Girls Math Olympiad
Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions:

(i) $a,b\in\{1,2,\ldots,2013\}$;

(ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.
2013 China Girls Math Olympiad
For any given positive numbers $a_1,a_2,\ldots,a_n$, prove that there exist positive numbers $x_1,x_2,\ldots,x_n$ satisfying  $\sum_{i=1}^n x_i=1$, such that for any positive numbers $y_1,y_2,\ldots,y_n$  with  $\sum_{i=1}^n y_i=1$, the inequality  $\sum_{i=1}^n \frac{a_ix_i}{x_i+y_i}\ge \frac{1}{2}\sum_{i=1}^n a_i$ holds.
2013 China Girls Math Olympiad
Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any  $x,y\in S$  there exists  $z\in S$  satisfying $x+y \equiv 2z \pmod{99}$. Determine all possible values of $m$.
2013 China Girls Math Olympiad
As shown in the figure, $\odot O_1$ and $\odot O_2$ touches each other externally at a point $T$, quadrilateral $ABCD$ is inscribed in $\odot O_1$, and the lines $DA$, $CB$ are tangent to $\odot O_2$ at points $E$ and $F$ respectively.  Line $BN$ bisects $\angle ABF$ and meets segment $EF$ at $N$.  Line $FT$ meets the arc $\widehat{AT}$ (not passing through the point $B$) at another point $M$ different from $A$.  Prove that $M$ is the circumcenter of $\triangle BCN$.
2013 China Girls Math Olympiad
Let $n$ ($\ge 4$) be an even integer.  We label $n$ pairwise distinct real numbers arbitrarily on the $n$ vertices of a regular $n$-gon, and label the $n$ sides clockwise as $e_1, e_2, \ldots, e_n$.  A side is called positive if the numbers on both endpoints are increasing in clockwise direction. An unordered pair of distinct sides $\left\{ e_i,e_j \right\}$ is called alternating if it satisfies both conditions:



(i) $2 \mid (i+j)$; and



(ii) if one rearranges the four numbers on the vertices of these two sides $e_i$ and $e_j$ in increasing order $a < b < c < d$, then $a$ and $c$ are the numbers on the two endpoints of one of sides $e_i$ or $e_j$.



Prove that the number of alternating pairs of sides and the number of positive sides are of different parity.
2012 China Girls Math Olympiad
Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that

$\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$
2012 China Girls Math Olympiad
Circles $Q_1$ and $Q_2$ are tangent to each other externally at $T$. Points $A$ and $E$ are on $Q_1$, lines $AB$ and $DE$ are tangent to $Q_2$ at $B$ and $D$, respectively, lines $AE$ and $BD$ meet at point $P$. Prove that

(1) $\frac{AB}{AT}=\frac{ED}{ET}$;

(2) $\angle ATP + \angle ETP = 180^{\circ}$.

[asy]import graph; size(5.97cm); real lsf=0.5; pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-6,xmax=5.94,ymin=-3.19,ymax=3.43; 

pair Q_1=(-2.5,-0.5), T=(-1.5,-0.5), Q_2=(0.5,-0.5), A=(-2.09,0.41), B=(-0.42,1.28), D=(-0.2,-2.37), P=(-0.52,2.96); 

D(CR(Q_1,1)); D(CR(Q_2,2)); D(A--B); D((-3.13,-1.27)--D); D(P--(-3.13,-1.27)); D(P--D); D(T--(-3.13,-1.27)); D(T--A); D(T--P); 

D(Q_1); MP("Q_1",(-2.46,-0.44),NE*lsf); D(T); MP("T",(-1.46,-0.44),NE*lsf); D(Q_2); MP("Q_2",(0.54,-0.44),NE*lsf); D(A); MP("A",(-2.22,0.58),NE*lsf); D(B); MP("B",(-0.35,1.45),NE*lsf); D((-3.13,-1.27)); MP("E",(-3.52,-1.62),NE*lsf); D(D); MP("D",(-0.17,-2.31),NE*lsf); D(P); MP("P",(-0.47,3.02),NE*lsf); 

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
2012 China Girls Math Olympiad
Find all pairs $(a,b)$ of integers satisfying: there exists an integer $d \ge 2$ such that $a^n + b^n +1$ is divisible by $d$ for all positive integers $n$.
2012 China Girls Math Olympiad
There is a stone at each vertex of a given regular $13$-gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the $13$-gon.
2012 China Girls Math Olympiad
As shown in the figure below, the in-circle of $ABC$ is tangent to sides $AB$ and $AC$ at $D$ and $E$ respectively, and $O$ is the circumcenter of $BCI$. Prove that $\angle ODB = \angle OEC$.

[asy]import graph; size(5.55cm); pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-5.76,xmax=4.8,ymin=-3.69,ymax=3.71; 

pen zzttqq=rgb(0.6,0.2,0), wwwwqq=rgb(0.4,0.4,0), qqwuqq=rgb(0,0.39,0); 

pair A=(-2,2.5), B=(-3,-1.5), C=(2,-1.5), I=(-1.27,-0.15), D=(-2.58,0.18), O=(-0.5,-2.92); 

D(A--B--C--cycle,zzttqq); D(arc(D,0.25,-104.04,-56.12)--(-2.58,0.18)--cycle,qqwuqq); D(arc((-0.31,0.81),0.25,-92.92,-45)--(-0.31,0.81)--cycle,qqwuqq); 

D(A--B,zzttqq); D(B--C,zzttqq); D(C--A,zzttqq); D(CR(I,1.35),linewidth(1.2)+dotted+wwwwqq); D(CR(O,2.87),linetype("2 2")+blue); D(D--O); D((-0.31,0.81)--O); 

D(A); D(B);  D(C); D(I); D(D);  D((-0.31,0.81)); D(O); 

MP( "A", A, dir(110)); MP("B", B, dir(140)); D("C", C, dir(20)); D("D", D, dir(150)); D("E", (-0.31, 0.81), dir(60)); D("O", O, dir(290)); D("I", I, dir(100));

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
2012 China Girls Math Olympiad
There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$.
2012 China Girls Math Olympiad
Let $\{a_n\}$ be a sequence of nondecreasing positive integers such that $\textstyle\frac{r}{a_r} = k+1$ for some positive integers $k$ and $r$. Prove that there exists a positive integer $s$ such that $\textstyle\frac{s}{a_s} = k$.
2012 China Girls Math Olympiad
Find the number of integers $k$ in the set $\{0, 1, 2, \cdots, 2012\}$ such that $\binom{2012}{k}$ is a multiple of $2012$.
2011 China Girls Math Olympiad
Find all positive integers $n$ such that the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ has exactly $2011$ positive integer solutions $(x,y)$ where $x \leq y$.
2011 China Girls Math Olympiad
The diagonals $AC,BD$ of the quadrilateral $ABCD$ intersect at $E$. Let $M,N$ be the midpoints of $AB,CD$ respectively. Let the perpendicular bisectors of the segments $AB,CD$ meet at $F$. Suppose that $EF$ meets $BC,AD$ at $P,Q$ respectively. If $MF\cdot CD=NF\cdot AB$ and $DQ\cdot BP=AQ\cdot CP$, prove that $PQ\perp BC$.
2011 China Girls Math Olympiad
The positive reals $a,b,c,d$ satisfy $abcd=1$. Prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geqslant \frac{{25}}{4}$.