Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 China Girls Math Olympiad
A tennis tournament has $n>2$ players and any two players play one game against each other (ties are not allowed). After the game these players can be arranged in a circle, such that for any three players $A,B,C$, if $A,B$ are adjacent on the circle, then at least one of $A,B$ won against $C$. Find all possible values for $n$.
2011 China Girls Math Olympiad
A real number $\alpha \geq 0$ is given. Find the smallest $\lambda  = \lambda (\alpha ) > 0$,  such that for any complex numbers ${z_1},{z_2}$ and $0 \leq x \leq 1$,  if  $\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|$,  then  $\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \right|$.
2011 China Girls Math Olympiad
Do there exist positive integers $m,n$, such that $m^{20}+11^n$ is a square number?
2011 China Girls Math Olympiad
There are $n$ boxes ${B_1},{B_2},\ldots,{B_n}$ from left to right, and there are $n$ balls in these boxes. If there is at least $1$ ball in ${B_1}$, we can move one to ${B_2}$.  If there is at least $1$ ball in ${B_n}$, we can move one to ${B_{n - 1}}$. If there are at least $2$ balls in ${B_k}$, $2 \leq k \leq n - 1$ we can move one to ${B_{k - 1}}$, and one to ${B_{k + 1}}$. Prove that, for any arrangement of the $n$ balls, we can achieve that each box has one ball in it.
2011 China Girls Math Olympiad
The $A$-excircle $(O)$ of $\triangle ABC$ touches $BC$ at $M$. The points $D,E$ lie on the sides $AB,AC$ respectively such that $DE\parallel BC$. The incircle $(O_1)$ of $\triangle ADE$ touches $DE$ at $N$. If $BO_1\cap DO=F$ and $CO_1\cap EO=G$, prove that the midpoint of $FG$ lies on $MN$.
2010 China Girls Math Olympiad
Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of

$$\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}$$

Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$
2010 China Girls Math Olympiad
In triangle $ABC$, $AB = AC$. Point $D$ is the midpoint of side $BC$. Point $E$ lies outside the triangle $ABC$ such that $CE \perp AB$ and $BE = BD$. Let $M$ be the midpoint of segment $BE$. Point $F$ lies on the minor arc $\widehat{AD}$ of the circumcircle of triangle $ABD$ such that $MF \perp BE$. Prove that $ED \perp FD.$



[asy]

defaultpen(fontsize(10)); size(6cm);

pair A = (3,10), B = (0,0), C = (6,0), D = (3,0), E = intersectionpoints( Circle(B, 3), C--(C+100*dir(B--A)*dir(90)) )[1], M = midpoint(B--E), F = intersectionpoints(M--(M+50*dir(E--B)*dir(90)), circumcircle(A,B,D))[0];

dot(A^^B^^C^^D^^E^^M^^F);

draw(B--C--A--B--E--D--F--M^^circumcircle(A,B,D));

pair point = extension(M,F,A,D);

pair[] p={A,B,C,D,E,F,M};

string s = "A,B,C,D,E,F,M";

int size = p.length;

real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;}

d[4] = -50;

string[] k= split(s,",");

for(int i = 0;i<p.length;++i) {

label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i]));

}[/asy]
2010 China Girls Math Olympiad
Prove that for every given positive integer $n$, there exists a prime $p$ and an integer $m$ such that

$(a)$ $p \equiv 5 \pmod 6$

$(b)$ $p \nmid n$

$(c)$ $n \equiv m^3 \pmod p$
2010 China Girls Math Olympiad
Let $x_1,x_2,\cdots,x_n$ be real numbers with $x_1^2+x_2^2+\cdots+x_n^2=1$. Prove that

$$\sum_{k=1}^{n}\left(1-\dfrac{k}{{\displaystyle \sum_{i=1}^{n} ix_i^2}}\right)^2 \cdot \dfrac{x_k^2}{k} \leq \left(\dfrac{n-1}{n+1}\right)^2 \sum_{k=1}^{n} \dfrac{x_k^2}{k}$$

Determine when does the equality hold?
2010 China Girls Math Olympiad
Let $f(x)$ and $g(x)$ be strictly increasing linear functions from $\mathbb R $ to $\mathbb R $ such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that for any real number $x$, $f(x)-g(x)$ is an integer.
2010 China Girls Math Olympiad
In acute triangle $ABC$, $AB > AC$. Let $M$ be the midpoint of side $BC$. The exterior angle bisector of $\widehat{BAC}$ meet ray $BC$ at $P$. Point $K$ and $F$ lie on line $PA$ such that $MF \perp BC$ and $MK \perp PA$. Prove that $BC^2 = 4 PF \cdot  AK$.



[asy]

defaultpen(fontsize(10)); size(7cm);

pair A = (4.6,4), B = (0,0), C = (5,0), M = midpoint(B--C), I = incenter(A,B,C), P = extension(A, A+dir(I--A)*dir(-90), B,C), K = foot(M,A,P), F = extension(M, (M.x, M.x+1), A,P);

draw(K--M--F--P--B--A--C);

pair point = I;

pair[] p={A,B,C,M,P,F,K};

string s = "A,B,C,M,P,F,K";

int size = p.length;

real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;}

string[] k= split(s,",");

for(int i = 0;i<p.length;++i) {

label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i]));

}[/asy]
2010 China Girls Math Olympiad
For given integer $n \geq 3$, set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$. Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$, one of these number does not lie in between the other two numbers in every permutations $p_i$ ($1 \leq i \leq m$). (For example, in the permutation $(1, 3, 2, 4)$, $3$ lies in between $1$ and $4$, and $4$ does not lie in between $1$ and $2$.) Determine the maximum value of $m$.
2010 China Girls Math Olympiad
Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$, $a^2=m_2^2+n_2^2$, and $m_1-n_1=m_2-n_2.$
2009 China Girls Math Olympiad
Show that there are only finitely many triples $ (x,y,z)$ of positive integers satisfying the equation $ abc=2009(a+b+c).$
2009 China Girls Math Olympiad
Right triangle $ ABC,$ with $ \angle A=90^{\circ},$ is inscribed in circle $ \Gamma.$ Point $ E$ lies on the interior of arc $ {BC}$ (not containing $ A$) with $ EA>EC.$ Point $ F$ lies on ray $ EC$ with $ \angle EAC = \angle CAF.$ Segment $ BF$  meets $ \Gamma$ again at $ D$ (other than $ B$). Let $ O$ denote the circumcenter of triangle $ DEF.$ Prove that $ A,C,O$ are collinear.
2009 China Girls Math Olympiad
Let $ n$ be a given positive integer. In the coordinate set, consider the set of points $ \{P_{1},P_{2},...,P_{4n+1}\}=\{(x,y)|x,y\in \mathbb{Z}, xy=0, |x|\le n, |y|\le n\}.$



Determine the minimum of $ (P_{1}P_{2})^{2} + (P_{2}P_{3})^{2} +...+ (P_{4n}P_{4n+1})^{2} + (P_{4n+1}P_{1})^{2}.$
2009 China Girls Math Olympiad
Let $ n$ be an integer greater than $ 3.$ Points $ V_{1},V_{2},...,V_{n},$ with no three collinear, lie on a plane. Some of the segments $ V_{i}V_{j},$ with $ 1 \le i < j \le n,$ are constructed. Points  $ V_{i}$ and $ V_{j}$ are neighbors if  $ V_{i}V_{j}$ is constructed. Initially, chess pieces $ C_{1},C_{2},...,C_{n}$ are placed at points $ V_{1},V_{2},...,V_{n}$ (not necessarily in that order) with exactly one piece at each point. In a move, one can choose some of the $ n$ chess pieces, and simultaneously relocate each of the chosen piece from its current position to one of its neighboring positions such that after the move, exactly one chess piece is at each point and no two chess pieces have exchanged their positions. A set of constructed segments is called harmonic if for any initial positions of the chess pieces, each chess piece $ C_{i}(1 \le i \le n)$ is at the point $ V_{i}$ after a finite number of moves. Determine the minimum number of segments in a harmonic set.
2009 China Girls Math Olympiad
Let $ x,y,z$ be real numbers greater than or equal to $ 1.$ Prove that

$$ \prod(x^{2} - 2x + 2)\le (xyz)^{2} - 2xyz + 2.$$
2009 China Girls Math Olympiad
Circle $ \Gamma_{1},$ with radius $ r,$ is internally tangent to circle $ \Gamma_{2}$ at $ S.$ Chord $ AB$ of $ \Gamma_{2}$ is tangent to $ \Gamma_{1}$ at $ C.$ Let $ M$ be the midpoint of arc $ AB$ (not containing $ S$), and let $ N$ be the foot of the perpendicular from $ M$ to line $ AB.$ Prove that $ AC\cdot CB=2r\cdot MN.$
2009 China Girls Math Olympiad
On a $ 10 \times 10$ chessboard, some $ 4n$ unit squares are chosen to form a region $ \mathcal{R}.$ This region $ \mathcal{R}$ can be tiled by $ n$ $ 2 \times 2$ squares. This region $ \mathcal{R}$ can also be tiled by a combination of $ n$ pieces of the following types of shapes (see below, with rotations allowed).





Determine the value of $ n.$
2009 China Girls Math Olympiad
For a positive integer $ n,$ $ a_{n}=n\sqrt{5}- \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$
2008 China Girls Math Olympiad
(a) Determine if the set $ \{1,2,\ldots,96\}$ can be partitioned into 32 sets of equal size and equal sum.

(b) Determine if the set $ \{1,2,\ldots,99\}$ can be partitioned into 33 sets of equal size and equal sum.
2008 China Girls Math Olympiad
Let $ \varphi(x) = ax^3 + bx^2 + cx + d$ be a polynomial with real coefficients. Given that $ \varphi(x)$ has three positive real roots and that $ \varphi(0) < 0$, prove that

$$ 2b^3 + 9a^2d - 7abc \leq 0.

$$
2008 China Girls Math Olympiad
Determine the least real number $ a$ greater than $ 1$ such that for any point $ P$ in the interior of the square $ ABCD$, the area ratio between two of the triangles $ PAB$, $ PBC$, $ PCD$, $ PDA$ lies in the interval $ \left[\frac {1}{a},a\right]$.
2008 China Girls Math Olympiad
Equilateral triangles $ ABQ$, $ BCR$, $ CDS$, $ DAP$ are erected outside of the convex quadrilateral $ ABCD$. Let $ X$, $ Y$, $ Z$, $ W$ be the midpoints of the segments $ PQ$, $ QR$, $ RS$, $ SP$, respectively. Determine the maximum value of

$$ \frac {XZ+YW}{AC + BD}.

$$
2008 China Girls Math Olympiad
In convex quadrilateral $ ABCD$, $ AB = BC$ and $ AD = DC$. Point $ E$ lies on segment $ AB$ and point $ F$ lies on segment $ AD$ such that $ B$, $ E$, $ F$, $ D$ lie on a circle. Point $ P$ is such that triangles $ DPE$ and $ ADC$ are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point $ Q$ is such that triangles $ BQF$ and $ ABC$ are similar and the corresponding vertices are in the same orientation. Prove that points $ A$, $ P$, $ Q$ are collinear.
2008 China Girls Math Olympiad
Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 - 7x_1)x_1^7 = 8$ and

$$ x_{k + 1}x_{k - 1} - x_k^2 = \frac {x_{k - 1}^8 - x_k^8}{x_k^7x_{k - 1}^7} \text{ for }k = 2,3,\ldots

$$



Determine real number $ a$ such that if $ x_1 > a$, then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$, then the sequence is not monotonic.
2008 China Girls Math Olympiad
On a given $ 2008 \times 2008$ chessboard, each unit square is colored in a different color. Every unit square is filled with one of the letters C, G, M, O. The resulting board is called harmonic if every $ 2 \times 2$ subsquare contains all four different letters. How many harmonic boards are there?
2008 China Girls Math Olympiad
For positive integers $ n$, $ f_n = \lfloor2^n\sqrt {2008}\rfloor + \lfloor2^n\sqrt {2009}\rfloor$. Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence $ f_1,f_2,\ldots$.
2007 China Girls Math Olympiad
A positive integer $ m$ is called good if there is a positive integer $ n$ such that $ m$ is the quotient of $ n$ by the number of positive integer divisors of $ n$ (including $ 1$ and $ n$ itself). Prove that $ 1, 2, \ldots, 17$ are good numbers and that $ 18$ is not a good number.
2007 China Girls Math Olympiad
Let $ ABC$ be an acute triangle. Points $ D$, $ E$, and $ F$ lie on segments $ BC$, $ CA$, and $ AB$, respectively, and each of the three segments $ AD$, $ BE$, and $ CF$ contains the circumcenter of $ ABC$. Prove that if any two of the ratios $ \frac{BD}{DC}$, $ \frac{CE}{EA}$, $ \frac{AF}{FB}$, $ \frac{BF}{FA}$, $ \frac{AE}{EC}$, $ \frac{CD}{DB}$ are integers, then triangle $ ABC$ is isosceles.
2007 China Girls Math Olympiad
Let $ n$ be an integer greater than $ 3$, and let $ a_1, a_2, \cdots, a_n$ be non-negative real numbers with $ a_1 + a_2 + \cdots + a_n = 2$. Determine the minimum value of $$ \frac{a_1}{a_2^2 + 1}+ \frac{a_2}{a^2_3 + 1}+ \cdots + \frac{a_n}{a^2_1 + 1}.$$
2007 China Girls Math Olympiad
The set $ S$ consists of $ n > 2$ points in the plane. The set $ P$ consists of $ m$ lines in the plane such that every line in $ P$ is an axis of symmetry for $ S$. Prove that $ m\leq n$, and determine when equality holds.
2007 China Girls Math Olympiad
Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
2007 China Girls Math Olympiad
For $ a,b,c\geq 0$ with $ a+b+c=1$, prove that



$ \sqrt{a+\frac{(b-c)^2}{4}}+\sqrt{b}+\sqrt{c}\leq \sqrt{3}$
2007 China Girls Math Olympiad
Let $ a$, $ b$, $ c$ be integers each with absolute value less than or equal to $ 10$. The cubic polynomial $ f(x) = x^3 + ax^2 + bx + c$ satisfies the property

$$ \Big|f\left(2 + \sqrt 3\right)\Big| < 0.0001.

$$

Determine if $ 2 + \sqrt 3$ is a root of $ f$.
2007 China Girls Math Olympiad
In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets $ 1$ point and the loser gets $ 0$ points. If the game is tied, each player gets $ 0.5$ points. Given a positive integer $ m$, a tournament is said to have property $ P(m)$ if the following holds: among every set $ S$ of $ m$ players, there is one player who won all her games against the other $ m-1$ players in $ S$ and one player who lost all her games against the other $ m - 1$ players in $ S$. For a given integer $ m \ge 4$, determine the minimum value of $ n$ (as a function of $ m$) such that the following holds: in every $ n$-player round robin chess tournament with property $ P(m)$, the final scores of the $ n$ players are all distinct.
2006 China Girls Math Olympiad
Let $a>0$, the function $f: (0,+\infty) \to R$ satisfies $f(a)=1$, if for any positive reals $x$ and $y$, there is $$f(x)f(y)+f \left( \frac{a}{x}\right)f \left( \frac{a}{y}\right) =2f(xy)$$ then prove that $f(x)$ is a constant.
2006 China Girls Math Olympiad
Let $O$ be the intersection of the diagonals of convex quadrilateral $ABCD$. The circumcircles of $\triangle{OAD}$ and $\triangle{OBC}$ meet at $O$ and $M$. Line $OM$ meets the circumcircles of $\triangle{OAB}$ and $\triangle{OCD}$ at $T$ and $S$ respectively.



Prove that $M$ is the midpoint of $ST$.
2006 China Girls Math Olympiad
Show that for any $i=1,2,3$, there exist infinity many positive integer $n$, such that among $n$, $n+2$ and $n+28$, there are exactly $i$ terms that can be expressed as the sum of the cubes of three positive integers.
2006 China Girls Math Olympiad
$8$ people participate in a party.



(1) Among any $5$ people there are $3$ who pairwise know each other.  Prove that there are $4$ people who paiwise know each other.



(2) If Among any $6$ people there are $3$ who pairwise know each other, then can we find $4$ people who pairwise know each other?
2006 China Girls Math Olympiad
The set $S = \{ (a,b) \mid 1 \leq a, b \leq 5, a,b \in \mathbb{Z}\}$ be a set of points in the plane with integeral coordinates. $T$ is another set of points with integeral coordinates in the plane. If for any point $P \in S$, there is always another point $Q \in T$, $P \neq Q$, such that there is no other integeral points on segment $PQ$.  Find the least value of the number of elements of $T$.
2006 China Girls Math Olympiad
Let $M= \{ 1, 2, \cdots, 19 \}$ and $A = \{ a_{1}, a_{2}, \cdots, a_{k}\}\subseteq M$. Find the least $k$ so that for any $b \in M$, there exist $a_{i}, a_{j}\in A$, satisfying $b=a_{i}$ or $b=a_{i}\pm a_{i}$ ($a_{i}$ and $a_{j}$ do not have to be different) .
2006 China Girls Math Olympiad
Given that $x_{i}>0$, $i = 1, 2, \cdots, n$, $k \geq 1$. Show that: $$\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}$$
2006 China Girls Math Olympiad
Let $p$ be a prime number that is greater than $3$. Show that there exist some integers $a_{1}, a_{2}, \cdots a_{k}$ that satisfy: $$-\frac{p}{2}< a_{1}< a_{2}< \cdots <a_{k}< \frac{p}{2}$$ making the product: $$\frac{p-a_{1}}{|a_{1}|}\cdot \frac{p-a_{2}}{|a_{2}|}\cdots \frac{p-a_{k}}{|a_{k}|}$$ equals to $3^{m}$ where $m$ is a positive integer.
2005 China Girls Math Olympiad
As shown in the following figure, point $ P$ lies on the circumcicle of triangle $ ABC.$ Lines $ AB$ and $ CP$ meet at $ E,$ and lines $ AC$ and $ BP$ meet at $ F.$ The perpendicular bisector of line segment $ AB$ meets line segment $ AC$ at $ K,$ and the perpendicular bisector of line segment $ AC$ meets line segment $ AB$ at $ J.$ Prove that



$$ \left(\frac{CE}{BF} \right)^2 = \frac{AJ \cdot JE}{AK \cdot KF}.$$
2005 China Girls Math Olympiad
Find all ordered triples $ (x, y, z)$ of real numbers such that



$$ 5 \left(x +  \frac{1}{x} \right) = 12 \left(y +  \frac{1}{y} \right) = 13 \left(z +  \frac{1}{z} \right),$$



and $$ xy + yz + zy = 1.$$
2005 China Girls Math Olympiad
Determine if there exists a convex polyhedron such that



(1) it has 12 edges, 6 faces and 8 vertices;



(2) it has 4 faces with each pair of them sharing a common edge of the polyhedron.
2005 China Girls Math Olympiad
Determine all positive real numbers $ a$ such that there exists a positive integer $ n$ and sets $ A_1, A_2, \ldots, A_n$ satisfying the following conditions:



(1) every set $ A_i$ has infinitely many elements;

(2) every pair of distinct sets $ A_i$ and $ A_j$ do not share any common element

(3) the union of sets $ A_1, A_2, \ldots, A_n$ is the set of all integers;

(4) for every set $ A_i,$ the positive difference of any pair of elements in $ A_i$ is at least $ a^i.$
2005 China Girls Math Olympiad
Let $ x$ and $ y$ be positive real numbers with $ x^3 + y^3 = x - y.$ Prove that $$ x^2 + 4y^2 < 1.$$
2005 China Girls Math Olympiad
An integer $ n$ is called good if there are $ n \geq 3$ lattice points $ P_1, P_2, \ldots, P_n$ in the coordinate plane satisfying the following conditions: If line segment $ P_iP_j$ has a rational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have irrational lengths; and if line segment $ P_iP_j$ has an irrational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have rational lengths.



(1) Determine the minimum good number.



(2) Determine if 2005 is a good number. (A point in the coordinate plane is a lattice point if both of its coordinate are integers.)
2005 China Girls Math Olympiad
Let $ m$ and $ n$ be positive integers with $ m > n \geq 2.$ Set $ S = \{1, 2, \ldots, m\},$ and $ T = \{a_l, a_2, \ldots, a_n\}$ is a subset of S such that every number in $ S$ is not divisible by any two distinct numbers in $ T.$ Prove that

$$ \sum^n_{i = 1} \frac {1}{a_i} < \frac {m + n}{m}.

$$
2005 China Girls Math Olympiad
Given an $ a \times b$ rectangle with $ a > b > 0,$ determine the minimum side of a square that covers the rectangle. (A square covers the rectangle if each point in the rectangle lies inside the square.)
2004 China Girls Math Olympiad
We say a positive integer $ n$ is good if there exists a permutation $ a_1, a_2, \ldots, a_n$ of $ 1, 2, \ldots, n$ such that $ k + a_k$ is perfect square for all $ 1\le k\le n$. Determine all the good numbers in the set $ \{11, 13, 15, 17, 19\}$.
2004 China Girls Math Olympiad
Let $ a, b, c$ be positive reals.  Find the smallest value of

$$ \frac {a + 3c}{a + 2b + c} + \frac {4b}{a + b + 2c} - \frac {8c}{a + b + 3c}.

$$
2004 China Girls Math Olympiad
Let $ ABC$ be an obtuse inscribed in a circle of radius $ 1$.  Prove that $ \triangle ABC$ can be covered by an isosceles right-angled triangle with hypotenuse of length $ \sqrt {2} + 1$.
2004 China Girls Math Olympiad
A deck of $ 32$ cards has $ 2$ different jokers each of which is numbered $ 0$. There are $ 10$ red cards numbered $ 1$ through $ 10$ and similarly for blue and green cards. One chooses a number of cards from the deck. If a card in hand is numbered $ k$, then the value of the card is $ 2^k$, and the value of the hand is sum of the values of the cards in hand. Determine the number of hands having the value $ 2004$.
2004 China Girls Math Olympiad
Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} + v\sqrt {wu} + w\sqrt {uv} \geq 1$. Find the smallest value of $ u + v + w$.
2004 China Girls Math Olympiad
Given an acute triangle $ABC$ with $O$ as its circumcenter.  Line $AO$ intersects $BC$ at $D$.  Points $E$, $F$ are on $AB$, $AC$ respectively such that $A$, $E$, $D$, $F$ are concyclic. Prove that the length of the projection of line segment $EF$ on side $BC$ does not depend on the positions of $E$ and $F$.
2004 China Girls Math Olympiad
Let $ p$ and $ q$ be two coprime positive integers, and $ n$ be a non-negative integer. Determine the number of integers that can be written in the form $ ip + jq$, where $ i$ and $ j$ are non-negative integers with $ i + j \leq n$.
2004 China Girls Math Olympiad
When the unit squares at the four corners are removed from a three by three squares, the resulting shape is called a cross. What is the maximum number of non-overlapping crosses placed within the boundary of a $ 10\times 11$ chessboard? (Each cross covers exactly five unit squares on the board.)
2003 China Girls Math Olympiad
Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} = x,$ $ \frac {AE}{AC} = y,$ $ \frac {DF}{DE} = z.$ Prove that



(1) $ S_{\triangle BDF} = (1 - x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} = x(1 - y) (1 - z)S_{\triangle ABC};$



(2) $ \sqrt [3]{S_{\triangle BDF}} + \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$
2003 China Girls Math Olympiad
There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$-th row and $ j$-th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say that the student is moved by $ [a, b] = [i - m, j - n]$ and define the position value of the student as $ a+b.$ Let $ S$ denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of $ S.$
2003 China Girls Math Olympiad
As shown in the figure, quadrilateral $ ABCD$ is inscribed in a circle with $ AC$ as its diameter, $ BD \perp AC,$ and $ E$ the intersection of $ AC$ and $ BD.$ Extend line segment $ DA$ and $ BA$ through $ A$ to $ F$ and $ G$ respectively, such that $ DG || BF.$ Extend $ GF$ to $ H$ such that $ CH \perp GH.$ Prove that points $ B, E, F$ and $ H$ lie on one circle.



[asy]

defaultpen(linewidth(0.8)+fontsize(10));size(150);

real a=4, b=6.5, c=9, d=a*c/b, g=14, f=sqrt(a^2+b^2)*sqrt(a^2+d^2)/g;

pair E=origin, A=(0,a), B=(-b,0), C=(0,-c), D=(d,0), G=A+g*dir(B--A), F=A+f*dir(D--A), M=midpoint(G--C);

path c1=circumcircle(A,B,C), c2=Circle(M, abs(M-G));

pair Hf=F+10*dir(G--F), H=intersectionpoint(F--Hf, c2);

dot(A^^B^^C^^D^^E^^F^^G^^H);

draw(c1^^c2^^G--D--C--A--G--F--D--B--A^^F--H--C--B--F);

draw(H--B^^F--E^^G--C, linetype("2 2"));

pair point= E;

label("$A$", A, dir(point--A));

label("$B$", B, dir(point--B));

label("$C$", C, dir(point--C));

label("$D$", D, dir(point--D));

label("$F$", F, dir(point--F));

label("$G$", G, dir(point--G));

label("$H$", H, dir(point--H));

label("$E$", E, NE);[/asy]
2003 China Girls Math Olympiad
(1) Prove that there exist five nonnegative real numbers $ a, b, c, d$ and $ e$ with their sum equal to 1 such that for any arrangement of these numbers around a circle, there are always two neighboring numbers with their product not less than $ \frac{1}{9}.$



(2) Prove that for any five nonnegative real numbers with their sum equal to 1 , it is always possible to arrange them around a circle such that there are two neighboring numbers with their product not greater than $ \frac{1}{9}.$
2003 China Girls Math Olympiad
Let $ \{a_n\}^{\infty}_1$ be a sequence of real numbers such that $ a_1 = 2,$ and $$ a_{n+1} = a^2_n - a_n + 1, \forall n \in \mathbb{N}.$$ Prove that $$ 1 - \frac{1}{2003^{2003}} < \sum^{2003}_{i=1} \frac{1}{a_i} < 1.$$